Does the question:
"What would you like on your birthday?"
mean the same as:
"What would you like for your birthday?"
Thanks a lot for your help
"What would you like on your birthday?" suggests that whatever it is that will be given will be given on the actual day of the birthday.
To me, "on" is asking what you want that day: a party, your favorite food for dinner, etc.
"For" is about a gift.
I want a party on my birthday and a new video game for my birthday.