In the Name of God, the Most Gracious, the Most Merciful
Salam 'Alaykum to you all
Introduction:

Before delving into this topic, it's important to acknowledge that it can be perplexing and may lead some individuals to question or even leave Islam. However, through careful examination and research, one can conclude that there are no chronological errors involved. I hold the belief that within the Qur'an, certain aspects are deliberately left open to interpretation, providing what could be viewed as "easy exits" for those with malicious intent or those seeking justification to stray from the path of Islam. This particular topic falls into that category.
In this article, I aim to demonstrate that there are valid arguments suggesting the absence of error, and that this serves as one of the avenues through which individuals may choose to depart from Islam, should they so desire. God has not made the Qur'an in such a way that when somebody reads it, that he absolutely cannot find what seemingly looks like errors (be it in history, science, or anything else). He has made it so that it apparently all is accurate, but left minor issues open to interpretation.
We can conclude the following:

  1. The name of Mary's father is completely unknown in both history and the canonical Gospels. He is only mentioned as "Joachim" (a Greek name!) in the gospel of James, and this is an apocrypha (a gospel rejected by all Christians). Thus, his name remains unknown. The Qur'an can make any claim and it would still be valid from a historical and Christian perspective.
  2. If Mary had any siblings is also unknown, both from the canonical Gospels and apocrypha. The Qur'an can even here make any claim and it would still be valid from a historical and Christian perspective.
  3. There is no mention of any Miriam (Moses and Aaron's sister) related to Amram found in the earliest fragments of the Torah. It can be argued that she was a later fabrication concocted by the Masoretes to make it look like the Qur'an contains a chronological mistake. I haven't research all of the manuscripts (before the 6th century), but I have looked at the Codex Sinaiticus and the Codex Vaticanus and the chapters that mention Miriam are not present in these codexes, which would have made it very easy for them to fabricate whatever they so desired. I am a strong believer in that the Masoretes were fierce opponents of Islam and did what they did only to discredit Islam and justify the entire Bible, I have proven this through countless posts here on Reddit.
  4. The Qur'an does not conflate the stories or eras of Mary with that of Moses and Aaron. It rather does the exact opposite.

I have in the past written on this topic extensively, but notice later on that the rebuttal was flawed, hence why I have written a new one that accurately refutes this completely.

Let's begin the response.
1. Did the Qur'an think Mary and Jesus were from the same era as Moses and Aaron?

The straightforward response to this is negative. To substantiate this assertion, we would need to present a verse that unequivocally indicates such a notion prior to the revelation of Surah 3 (Al-Imran), as any post-Surah 3 content could be interpreted as an attempt to rectify an earlier error. In Surah 2:87, God states:
"And indeed, We gave Moses the book, and We caused to follow after him messengers. And We gave Jesus, son of Mary, clear proofs and supported him with the spirit of holiness. So is it that every time a messenger comes to you with what does not meet your desires, you become arrogant and a party you deny and a party you kill?" (2:87)
The phrase "We gave Moses the book, and We caused to follow after him messengers" explicitly indicates that the author of the Qur'an was aware of a succession of messengers between Moses and Jesus. The mention of Moses and Jesus in this context, along with the specific reference to a lineage of messengers between them, serves the sole purpose (and I genuinely believe this) of providing a clear response to the contention of chronological error raised in Surah 19:28, "O sister of Aaron."
In other words, why would the Qur'an be stating that there was a line of messengers sent between Moses and Jesus if not for this exact purpose?
2. Does Sura 3 narrate the stories of Moses and Aaron?

It does not, it does mention them by name but it also mentions Abraham and others by name, while it does not narrate any of their stories. If the author of the Qur'an really conflated the eras of Mary with that of Moses/Aaron, then why are their stories not narrated in this chapter? Why would an entire chapter be named after "Amram" the father of Moses and Aaron, yet fail to narrate their stories? Because it's not about Amram or Moses and Aaron. It's about Mary and her family (and father, whose name was 'Imran).
3. "House of Abraham" and "House of 'Imran" - Abraham and Amram? no!

This is one of the concerns that they raise against the Qur'an when it comes to this topic: Why are Moses and Aaron omitted in 3:33:
"God did choose Adam and Noah, the family of Abraham, and the family of 'Imran above all people-" (3:33)
And then they say "So did God mention the family of Abraham and 'Imran, Mary's father, and omitted Amram and hence omitted Moses and Aaron?"
And the simple answer to this is: No! God indeed included Moses and Aaron in "Family of Abraham," but this Chapter is about Mary's family, the family of 'Imran, hence the specific mention of his family being chosen, but this is also why God says in the very next verse:
"Offspring, one of the other: And God heareth and knoweth all things." (3:34)
Meaning, they are all descendants of one another.
In Old Testament Biblical times, God had chosen Abraham's family and that of Amram, but in later times, God had also chosen the family of 'Imran, and thus mentioned their virtue, and omitted the mention of Amram's family because they are inherently already included in the family of Abraham, and this chapter is not about them, but rather about 'Imran and his family, hence the specific mention of choosing them over all others.
4. How can we definitely know that Amram and 'Imran are two distinct names?

Amram is not mentioned in the Qur'an by name and neither is Jochebed. This chapter is about 'Imran and his family, a completely different person and a completely different name. The Arabic way of saying "Amram" would be عَمْرَم.
The name "عَمْرَم" (Amram) and "عمران" ('Imrān) are not the same in Arabic. They are two distinct phrases with different spellings, pronunciations, and meanings. "عَمْرَم" (Amram) is not even recognized as an Arabic name or word to begin with, while 'Imran is. 'Imran is derived from the Arabic root "ع م ر" (ʿ-m-r), which conveys the idea of flourishing or thriving. The Hebrew name "Amram" is derived from the Hebrew root "עָמַר" (amar). The Hebrew root "amar" and the Arabic root "'مر" ('mr) share some phonetic similarity, but are two distinct words in their respective languages with different meanings and usage.
In Hebrew, "עָמַר" (amar) typically means "to be exalted," "to be strong," or "to be lofty."
In Arabic, the root "'مر" ('mr) carries various meanings depending on context, including "to live," "to prolong," or "to command." This root is used in forming various Arabic words with different meanings and connotations.
'Imran is simply the Arabic version of what Mary's father was called, and none of us really knows what his name was in Hebrew. To say that the Qur'an mentions Amram as 'Imran is fundamentally inaccurate, and even I made this mistake in the past. I completely conflated the two names because of their phonetic similarity.
In short, yes, God omitted the specific mention of Amram's family in this chapter because this chapter was not about them. Is it the end of the world? Are we supposed to cry our eyes out and complain (much like the Jews are regarding this omission)? Of course not. God does whatever He wants and the omission is clearly clarifies in "Offspring, one of the other" (3:35), but that is not enough for them and that's just too bad. Womp womp... let's move on.
5. Why did God quote the Jews saying "O sister of Aaron"?

By doing so, He revealed the fabrications of the Greek gospel authors. They all concocted the character of Joseph, purportedly the husband of Mary and of Davidic lineage, thereby implying Jesus' Davidic heritage and fulfilling the prophecies of Jeremiah regarding the "Branch" of David. However, these prophecies pertained to a king, whereas Jesus was not recognized as a king. Despite this, they were so convinced that Jeremiah's prophecy referred to Jesus that they fabricated an entire persona and claimed him to be Mary's husband. Yet, Mary's virginity was a pivotal sign to the people. How could this remain a sign if Mary were married? Their actions diminished the significance of Mary's virtuous virginity and the miraculous nature of Jesus' birth. Why would God orchestrate the miraculous birth of Jesus only to allow Mary to simultaneously marry, providing grounds for the assumption that she and Joseph had a child?
Songs of Solomon ch. 6 is proof that they fabricated Joseph and hence made Jesus of Davidic lineage. It literally states that Mary was placed in the "chariots" of Amminadab, the father in law of Aaron, which would make her of the lineage of Aaron, because she descended from Amminadab, which also would make her a "sister" of Aaron (since they're in the same family).
6. 'Songs of Solomon' ch. 6 is a prophecy about Mary and Jesus, proving their lineage directly from Amminadab (instead of David):

Verses 2-3:

"My beloved has gone down to his garden, to the beds of spices, to browse in the gardens and to gather lilies."
"I am my beloved’s and
my beloved is mine,
He who shepherds his
flock among the lilies.”
This is mirroring Matthew 6:28-29:
“And why worry about your clothing? Look at the lilies of the field and how they grow. They don’t work or make their clothing, yet Solomon in all his glory was not dressed as beautifully as they are."
Jesus is metaphorically referred to as a shepherd in the Gospels, and here he is described as browsing in the garden shepherding his flock among the lilies. He also explicitly mentions the name of Solomon for some weird reason, and it is only now that we can see why he mentioned Solomon: To quite literally refer back to Songs of Solomon 6 and this prophecy of him and his mother. Can't get any clearer than this.
Verses 5-6:

"Turn your eyes from me; they overwhelm me. Your hair is like a flock of goats descending from Gilead. Your teeth are like a flock of sheep coming up from the washing. Each has its twin, not one of them is missing."
Mirroring Jesus as a shepherd. Luke 15:3-7:
"Then Jesus told them this parable: “Suppose one of you has a hundred sheep and loses one of them. Doesn’t he leave the ninety-nine in the open country and go after the lost sheep until he finds it?..."
During Jesus' time, Gilead was part of the broader region known as Perea, which was governed by Herod Antipas, son of Herod the Great. Jesus traveled throughout the regions of Judea and Perea during his time on earth, as mentioned in passages such as Luke 13:22.
The words "Descending" and "coming up" are mentioned here too and seem very cryptic, mirroring his ascent into heaven and his ascent, the second coming.
Also, read:
"When the son of man comes in his glory, and all the angels with him, he will sit on his glorious throne. All the nations will be gathered before him, and he will separate the people one from another as a shepherd separates the sheep from the goats." - Matthew 25:31-32 (NIV)
Again, very cryptic and only now do we understand why this was specifically said this way.
Verses 8-9:

"Sixty queens there may be, and eighty concubines, and virgins beyond number; My dove, my perfect one, is unique, the favorite of her mother, flawless to her who bore her. The maidens saw her and called her blessed; the queens and concubines also, and they praised her."
Mirroring the Virginity of Mary and her being morally innocent.
The word used here (that they've translated as "perfect one") is "תַמָּתִ֔י", defined like this in dictionaries:
Hebrew dictionary:
2 - a. complete, morally innocent, having integrity
Source: מקור: Open Scriptures on GitHub
Creator: יוצר: Based on the work of Larry Pierce at the Online Bible
Verse 11:

"I went down to the grove of nut trees to look at the new growth in the valley, to see if the vines had budded or the pomegranates were in bloom."
Mirroring the Qur'anic story of Mary:
"And she conceived him, and she withdrew with him to a remote place. And the pains of childbirth drove her to the trunk of a palm tree. She said, 'Oh, I wish I had died before this and was in oblivion, forgotten.' But he called her from below her, 'Do not grieve; your Lord has provided beneath you a stream. And shake toward you the trunk of the palm tree; it will drop upon you ripe, fresh dates. So eat and drink and be contented. And if you see from among humanity anyone, say, 'Indeed, I have vowed to the Most Merciful abstention, so I will not speak today to [any] man.'" (Quran 19:22-26)
This is what a 12th century classical Arabic dictionary said about "Nakhla Mahmoud" (نخلة محمود) i.e. the word "Nakhal" that is translated as "palm tree," and a place called "Mahmoud" near Mecca:
Phrase: "نخلة محمود": "A place in the Hijaz near Mecca, with palm trees and vineyards. It is the first station for those leaving from Mecca. In the marginal notes of Abu Musa: Imran al-Nakhlawi was from the belly of a palm tree, and his place was there, and then Sa'id bin Jamhan met him."
Source: Yāqūt al-Ḥamawī, Muʿjam al-Buldān (d. 1229)
The verse in Songs of Solomon uses the same word ("Nachal") in Hebrew, and it means the following according to Hebrew dictionaries:
Heb root: נַחַל (n-m)
2. PALM-TREE
Source: מקור: Open Scriptures on GitHub
Creator: יוצר: Based on the work of Larry Pierce at the Online Bible
And another dictionary:
[INDENT]n.[m.] perhaps palm-tree (Arabic نَخْلٌ, n. unit. نَخْلَةٌ; v. PerlesJQ, July, 1899, 688);— only pl. abs., כִּנְחָלִים נִטָּ֑יוּ Nu 24:6 like palm-trees, which are stretched out, spread out (as to foliage). So Perlesl.c., who compares כערבי נחל Ecclus 50:12,

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I have no idea why the website cut half the post off... how do I edit?