First of all, elections are overseen by each state individually. While there is Federal law involving elections, it’s up to the States to implement them. Due to the Electoral College, the Presidential Election is really a weighted combination of the results of 50 separate State elections (and DC). So ypu can argue that we really don’t have any national elections at all, so each state runs their own.
Then, there is another complicating factor that there is no one piece of Federal ID that everyone is mandated to have. The closest thing is the Social Security Card, but that isn’t really supposed to be used as ID. Not everyone has a passport, and there is no national ID card. The closest thing we have to a universal ID is the driver’s license, but again that is managed on a state-by-state basis.
The main argument here is that when someone registers to vote, they must submit proof of residence, but Federal law holds that they do not have to show proof of citizenship. They merely have to attest that they are a citizen, and lying on that form is a crime. Many states object to that. Some go as far as to say that if you do not bring your proof of citizenship whe you register to vote, they will only let you vote in Federal elections, not state ones.
The fundamental question is: if you know you are a citizen but have lost both your passport and your birth certificate, should you be disenfranchised? Republicans clearly say “yes”.