For the most part - it works exactly like what you described. What kinds of ID are valid, and to some extent whether you are required to present one at all, depends on which state you live in.
The fake mustache double voter would have to know the details of another person who is already registered to vote (only some states allow same-day voter registration) and gambles on the other person not showing up to vote.
One big difference between the US and a lot of other democracies (when it comes to voting laws) is that the US doesn’t have any form of universal national identification documents - pretty much everything is issued on a state-by-state basis, and with very few exceptions those state level IDs don’t actually say anything about citizenship - noncitizen permanent residents are allowed to get driver’s licenses.
There isn’t one centralized database with that information for everyone. Each state, and even inside one state each county, maintains their own records, so someone who is born in Florida, gets married in California, and then has a kid born in Oregon would need to contact each of those states for proof of those events if they lost the original copies for whatever reason. There is a national system that can (mostly) check against those state and local level records, but it also has limitations. Passports and birth certificates can be proof of citizenship, but don’t prove where you live at the time of the election even if you do have one on hand.
The voter registration process already gives the state enough information to determine whether an individual is eligible to vote in that state. For federal elections (which are still run by the states, just for federal level positions), that includes confirming that the person is a citizen, but each state is able to decide whether to allow noncitizen residents to vote on local government issues or not.