Yes and No. There is no specific text or teaching in Islam that says that one must or must not celebrate birthdays.
But that said the majority of Muslims; especially non-western Muslims tend to not celebrate birthdays. The reasons for this are many, ranging from it not being part of their culture to economics of funding an extra unnecessary occasion/festival.
I as the author of this answer reside in the United Kingdom. Like most Muslim Families we are very close and I have over 150 relatives that I am always in touch with… these are not my friends or colleagues but just family alone.
Now if this culture was prevalent in my family. I would have a birthday party every other day to attend. I would have to take hours out of my time attending these parties. Add to that the hours looking for a suitable unique present that I’ve not already gifted to any of my other 149 relatives over the past years. Then not to forget that even if I was to spend a bare minimum of £10 per gift, per person. That would add to a colossal deficit to my annual expenditure.
On top of that expense, add the financial loss of me being away from my business and Muslim women are no different than any other ladies. In that, they too wouldn’t be seen dead in the same outfit twice. So all us married Muslims would have to arrange 150 dresses per year for our wives. Last but not least I’d have to arrange about 6 parties a year for my own little clan.
Non-Muslim westerners can get away with birthdays due to the unfortunate reality of that they have altogether cut their ties of kinship. Thus often only have 4-7 family members celebrating birthdays. The financial burden is far less because of this and the fact that they use this occasion to bond with the little family members that they do keep in touch with.
Those Muslims that have already begun celebrating birthdays, have started the trend often due to their effort to bridge cultural differences or to relieve pressure of cultural differences. They often start as very little affairs, with the parents celebrating the children’s birthdays only. Dragging into the mix close aunties, grandparents and only the parents giving the gifts. But every year I have seen with these families the whole thing growing and becoming the nightmare I described above.
With the eventual result being that expectation of presents will rise year upon year. Presents from all attendees will be expected and competition amongst parents to out shine the Ahmeds (Joneses) will become the norm.