1.NounA cake, often iced and decorated, made to celebrate a person's birthday.
2.NounIn particular, a buttery, frosted vanilla cake of the sort commonly made to celebrate birthdays in the US, used in compound words to attribute a buttery or creamy, sugary, vanilla cake flavour or aroma to other things.
"Pat crushed up a package of birthday cake Oreos and sprinkled them over birthday cake ice cream."