Logistically speaking, how would a person be born on a certain day, but to uncertain parents? I assume that if it was on a certain day, then people would have been around to see the actual birth happening and remember that day, but in that case, to what extent could the identity of the mother be unknown?

the mother died while giving birth and no one knew her name?

The birth something can be divined by sorcery. This is how they know that Direnni Tower was made 2500 BC.

Or the baby was dropped off somewhere immediately following birth, anonymously. You can tell when a baby's just been born. Head shape, and all.

It would be something like this because they need the hero as far away from a normal life as possible so that he can fulfill the prophecy.

Uncertain to prophesy doesn't mean uncertain to record-keeping. The prophesy could have prophesied that a hero would be born on March 28th, but it didn't know which one of the kids born on March 28th of any year would end up being the hero. The hero's parents could have given him of to other people to raise, telling the adoptive parents the kid's birthday but not their own names.