The Google hits were meant by way of illustration, as my many years of dealing with my mother tongue are hard to present as evidence. I'd normally say "birthday present", but "birthday gift" is fine, too. I agree that is used more in the US, these days.
gift /ft/ noun, verb
1 a thing that you give to sb, especially on a special occasion or to say thank you SYN: present: The watch was a gift from my mother. http://www.oup.com/oald-bin/web_getald7index1a.pl
"If she had known what birthday gift was coming every minute nearer and nearer, she would not have said to herself ..."
Little Women, by Louisa May Alcott
"Silly Milly's birthday gift. Only five she was then. "
Ulysses, by James Joyce
"I have a birthday gift for you. Do you want it?"
Household Words, by Charles Dickens