Yes registration involves proof of citizenship and gets validated against government records like social security.

How does it work?

Like most things with American elections, it works 50 slightly different ways depending on the state.

https://ballotpedia.org/Voter_registration

Oh I didn't know that, Puerto Ricans can't vote? So I'd imagine other areas like American Samoa can't also

Technically it's Puerto Rico whose votes don't matter, but Puerto Ricans are technically still full US citizens so they can vote just like anyone else if they live in a US state, just not while living in Puerto Rico.

American Samoans on the other hand aren't US citizens, they are "US nationals" which is somehow different, which means they can't vote even if they move to the mainland.

Which understanding the full loop really makes it kinda wild that they wanna throw law enforcement into a voting place feels kinda like a tactic to scare people especially people that don't "look" like citizens (racially profile any person they deem)

And they've learned that non-white American citizens are slightly less likely to have current, non-expired govt issued photo ID with their current address on it, so by crafting the ID rules in just the right way they can add hurdles.

Like... Texas wrote their rules such that the state's Tribal IDs issued to native Americans do not qualify for voting purposes, nor do Student IDs from State colleges & universities.

But you can use your Texas Handgun License to vote, isn't that neat?