I would tend to assume that in each state, the law would require
Your assumption would be incorrect. In many states (probably most U.S. states), a representation of the person registering to vote that they are a U.S. citizen is sufficient to be validly registered to vote.
Other information could be called upon to prove citizenship if the eligibility of the voter to vote based upon citizenship is formally disputed.
In practice, the number of cases of non-citizens voting or registering to vote is infinitesimal, and the cases that do arise mostly involve good faith mistakes lawful permanent residents (i.e. green card holders) who are in the process of becoming naturalized U.S. citizens, who misunderstood where they were administratively in the naturalization process, and become U.S. citizens shortly after registering to vote or voting.