The name is on the list, so the person can vote. Why would you need them to show an id for that? You would need to establish the identity first (which everybody would have anyways, should the US not be a bunch of third world countries in a trench coat), but not eligibility.
So all you need to do is know somebody's name for that voting station. And since we're not checking IDs, when the "right" person shows up, how do we know they're the right person?
I have to show ID to get into my local zoo, but not to vote someone onto the board in charge of the zoo. That doesn't make sense.