Most of those voter registration services were non-partisan anyways, and may have ended up helping Bush by getting more low information voters to the polls. There were of course a few GOP voter registration outfits pretending to be non-partisan that would shred any Democratic registrations, but that's a whole other topic.
Anyways, it's the responsibility of the county/state to make sure that the person attempting to register is an adult citizen.
Without some kind of voter ID requirement...how?
During the registration phase, the government typically has to process the registration. Presumably, that involves making sure the potential registree actually exists and is a citizen, neither of which should require an ID card.