We used to look people up in big books, that are printed for the occasion, to make sure their piece is valid and they haven’t already voted - which took a long time and led to queues. That’s the part we digitized.
But you can vote without ID as long as you have that piece of paper and know your birthdate.
I’m actually not sure how the homeless vote (they can’t receive the piece of paper), but I guess they probably do a mail vote at their local municipality.