You have to check school policy first, usually at the beginning of school year, they will distribute a note saying how student's birthday is treated. In my girl's case, they don't allow any party food at all, all they ask is parents can give teacher a notice about the upcoming birthday of child, then on that day, the teacher will give the child a special hat to wear and the class will sing song for him/her, and your child get a small gift (pen, crayon, eraser, etc..).
So I'd say it varies by school....