Allah says,
“Call them (adopted sons) by (the names of) their fathers, that is more just with Allaah…” [al-Ahzaab 33:5].
As it is in this world, so it will also be in the Hereafter, as the Prophet (peace and blessings of Allaah be upon him) said:
“On the Day of Resurrection, each betrayer will have a banner raised beside him, and it will be said, this is the betrayer of So and so the son of So and so.” (Narrated by al-Bukhaari, 5709, and Muslim, 3265).
Shaykh Bakr Abu Zayd (may Allaah preserve him) said: This is one of the beauties of sharee’ah, because calling a person by his father’s name is more appropriate for knowing who is who and telling people apart. The father is the protector and maintainer of the child and his mother both inside and outside the home.
This is why the father mixes with people in the marketplaces and takes risks by travelling to earn a halaal living and strive for their sakes. So the child is given the name of the father, not of the mother who is hidden away and who is one of those whom Allaah commanded (interpretation of the meaning):
“And stay in your houses…” [al-Ahzaab 33:33]
(Tasmiyat al-Mawlood, 30, 31).
On the basis of the above, there is no blood tie between the husband and wife, so how can she take his surname as if she is part of the same lineage? Moreover, she may get divorced, or her husband may die, and she may marry another man. Will she keep changing her surname every time she marries another man? Furthermore, there are rulings attached to her being named after her father, which have to do with inheritance, spending and who is a mahram, etc. Taking her husband’s surname overlooks all that. The husband is named after his own father, and what does she have to do with the lineage of her husband’s father? This goes against common sense and true facts. The husband has nothing that makes him better than his wife so that she should take his surname, whilst he takes his father’s name.
Hence everyone who has gone against this and taken her husband’s name should put matters right. We ask Allaah to put all the affairs of the Muslims right.
(islamqa.com)