The characters combine logically: 生日 (shēngrì) means “birth day,” and 蛋糕 (dàngāo), literally “egg pastry,” means “cake.” Therefore, 生日蛋糕 (shēngrì dàngāo) is unambiguously a “birthday cake.”
While the practice of eating a sweet cake with candles for a birthday is a Western import, it has been widely adopted and adapted in China.
生日蛋糕 is a common term used in everyday life, from casual conversation to ordering at a bakery.