No. it does not matter how it is classified. Compassion has two different types of gifts: Child gifts and Family Gifts. (Actually three because you can give a Center gift as well if you call it in). I'm not sure birthdays are as big a deal in some of these countries than we make them because I rarely hear anything about them in letters. (Though the 15th birthday does seem particularly important, at least for girls, in many places.) You might send a separate birthday card/letter if you want to make it clear you are remembering their years on the earth.
Kayce, thank you so much for wishing to bless your child on her birthday! We actually have three types of gifts - general child gifts, birthday gifts, and family gifts. We have Christmas gifts as well. I just looked and the gift you gave is actually designated as a birthday gift. :)