Non-american here so wondering, can literally anyone in the US walk into a voting booth and cast a vote? If so... Why would you allow that? And why would you not want to prevent it?
I don't have to present any ID in order to vote. Since I am already on the voting rolls, all I do is show up at the poll, tell them who I am, and they find my name in the (now electronic) book. I sign it and then I can go vote.
Now, when I initially registered, I had to prove I was eligible. Most voter registration is done here in conjunction with getting your driver's license (or other non-driver state ID). Since you have to produce your birth certificate and/or naturalization papers, they can check your eligibility right there.
A lot of the misinformation from the Right stems from how closely the DMV is tied into voter registration. Non-citizens can drive, of course, and some states even give licenses to people here illegally. Republicans assume all these people get signed up to vote, also, but they do not. Only people who can prove their eligibility get registered.
There are voter rolls that each state maintains of eligible voters. These rolls have to conform to minimum guidelines, like allowing all US citizens without felonies 18 years or older to vote. However, the federal government doesn't have any never had any maximum guidelines.
If a state decided that all 16 year old citizens could vote, the Federal Government couldn't do anything. Same with how different states address allowing felons who have served their time the ability to vote. In theory, this could include allowing non-citizens the opportunity as well.
This was built into the Constitution as an option. All elections, including for President, have some mechanism in place so that it doesn't matter if a state has a lesser or greater percentage of its residents vote over other states. After all, the Constitution was written in mind where several states would have significant slave populations.
It is unknown if a federal maximum guideline would be constitutional, since it really messes with the division of powers between the states and the federal government.
Voting in person in the US usually involves a checkin process where an election worker looks up your name and address to 1. Confirm you are registered to vote, 2. Confirm you are in the correct location to vote, and 3. Confirm that you haven’t already voted.
After voting you follow a similar process to check out, so they have a record that you checked in, voted, and checked out.
Point 2 is because in larger towns & cities there are often multiple voting locations, and voting locations are based on where in the town you live. For example in my city I vote at a church two blocks away from my home. If I tried to vote at a different location in my city they wouldn’t allow it, and would tell me to go to that church. (I think I could cast a provisional ballot at an alternate location if I provided sufficient ID, but that vote would only be counted after the election ended and records showed I didn’t vote at my usual location.)