I know a few people that don't have an official birth date. My brother in law is Papuan and no one in his village has an official birth year or date since they just don't keep track of it, so when he got his first official government ID at a much later stage of his life, he chose an arbitrary date cause it means nothing to him. He's probably around 35-40, but his birthday says he's in his 60s.

It's not all that rare even in the West, especially for older people. Records get lost or never get collected in the first place.