Okay, so why would they call them Unbirths if they don't even know what they are?
You gave an explanation as to why the Unbirths are called what they are outside of the game. Now, all I'm asking is for a reason why in-game the Unbirths would be called what they are. I don't think they'd just pick a name out of the blue that has no relevance to something, there has to be a reason for their name.
Like, Heartless -> Called such due to their cruel and instinctive nature
Nobodies -> Called such due to them lacking an identity of their own, their identity taken up by their hearts