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.

You register to vote, are assigned a polling place where your name will appear on a list of registered voters, and you go to that polling place and tell them your name.

If you're trying to fake it, you need to know what address and name someone else is registered at, what polling place they were assigned, and you have to hope they don't show up to vote too.

All of these uncertainties mean it's pretty difficult for an individual to do any serious (if any!) voter fraud.

Remember we had voting for a long time before magnetic strips and plastic.