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.
2019년 2월 27일
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"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.