Traditionally, cakes carried the name or initial of the birthday person, as sung for centuries by kids in the rhyme:
Pat a cake, Pat a cake, baker's man
Bake me a cake as fast as you can
Pat it and prick it and mark it with a 'B'
And put it in the oven for Baby and me
One of those things you think 'that's how it's always been' but all those hundreds of years ago, it was the thing to do and then became a tradition... If anyone can find out who started it, you get a cupcake!
Wonder when we started putting people's ages on the cake too, or is that just me...