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.
That prompts an investigation. The “right” person casts an affidavit ballot and the police and courts investigate. If the count is close, the loser usually sure to recount and verify, and any of these incidents then become political kindling. It doesn’t happen because it isn’t worth it individually and difficult to coördinate en masse.
[^] except for the case of felony disenfranchisement laws, which I personally believe are a travesty
This can be argued for any hindrance to bureaucracy. On the balance you get a much more robust system, with fewer centralized fail-safes.
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.