I gift itself would be called a "Birthday Gift".
I bought Sam a birthday gift
I bought Nicola a Christmas gift
If you are talking about the gift and explaining who you bought it for you would then say birthday gift for (and the person's name).
This is a birthday gift for Sam
This is a Christmas gift for Nicola
If you are talking about who possesses the gift (the owner) then you would use (person's name)'s birthday gift.
This is Sam's birthday gift
This is Nicola's Christmas gift
You can also use the expression "gift for" to specify the occasion that the gift has been bought for.
This is a gift for Sam's birthday
This is a gift for Christmas (in this case Christmas isn't specific to a single person so you don't use the persons name)
Gift is more common in American English. Present is commonly used in British English.
I hope that helps, but please let me know if you have any other questions.