The writer used "for" because of the phrase "What do you want for your birthday?".
The answer would be "For my birthday, I want a bicycle"
So when he says "for our birthdays.." he means that he wants to receive something.
Your phrase, "on our birthdays" would mean that specifically on the day of their birthday, they would tell their families and friends the present they want to have(which would be too late, because everyone will have bought and brought a gift already ;) )
To make the sentence exact, the writer could indeed have used "For our birthday presents".
But he makes the sentence more elegant by assuming that we know they are talking about presents.