Assalamu alaikum wa rahmatullahi wa barakaatuh,
Although not a daleel for us, it is worth noting that even not all Christians celebrate birthdays. Jehovah's Witnesses and some others believe it wrong to celebrate birthdays, either of religious figures, their own, or their childrens'.
Firstly they say Jesus (as) didn't command them to celebrate birthdays, the bible doesn't command them to, and the early Christians never celebrated birthdays. As Muslims, the same goes for us, no example from our scriptures, our prophet (saws) or the sahabah (raa). This includes the prophets (saws) birthday, there is no example.
Secondly, birthdays are pagan in origin, according to all historical research. Some say Egyptian origins, some say Roman, some say earlier, but all agree the origin is from pagan beliefs. The birth date of pharoahs became a celebration for when they were considered to have transformed into gods. Horoscopes were cast on this day. In other societies there were also beliefs that evil spirits visited a person each year on the date he was born, thus the need for candles, and gathering to scare them away etc. Gifts would appease the evil spirits.
Jehovahs witnesses believe following practices rooted in pagan customs, rites, and origins, is forbidden to them (although they still have beliefs re: Jesus (as) that Islam would regard as shirk). As Muslims, this avoidance of pagan origin customs applies to Muslims more. How is it that non-Muslims are trying to avoid pagan practices more than Muslims? It should be the other way around!
Thirdly, is a year closer to death, less time to do good deeds something to celebrate as Muslims? Food for thought.
We so blindly get caught up in just following what most others around us do in society, viewing it as harmless and without giving it much thought...
Let us make the two eids our celebrations, and let us give gifts, express love, gather, try to make others happy, at Eid and also any and all times throughout the year, without singling out birthdays..