Indeed, you can hear very often – it was a birthday.
We focus on the word day. Day – masculine noun. Day (what did?) was.
It was a day – it sounds agreed. Say: it was a day – it does not sculpt. Disagreement in childbirth. It was – it, the middle gender.
Take the right thing – was a birthday.
I agree, you can often find the expression “there was a day” or “my day”. Although it would be the right “birthday”, “my birthday”.
In fact, everything is simple, if you make out this expression on the members of the sentence: the predicate (what did you do?) was, subject (what?) Day, Supplement (what?) birth.