생일 축하해 could be the short version of 생일인거 축하해, not just only of 생일을 축하해. But they all mean the same thing after all.

"생일(birthday)을(particle for the object of the sentence) 축하합니다(congratulation)." = "생일 축하합니다." It is common that the object marker '을' is skipped in conversation.