The check in the US is done when you register to vote. Once you are registered, a variety of proofs of ID can be used to vote at your polling location.
Requiring a passport and birth cert or some other strong ID are unnecessary at the actual voting site. The main reason for doing this is to make voting take longer, and be more strenuous, which means that you can have a greater effect on election results by manipulating the number of polling stations for an area.