No, the US does not have a database of citizens or a national ID. There's been long-standing resistance to such a thing, as part of the general American distrust of centralized government and what a tyrannical government could do with that information. That resistance has even been bipartisan, coming more from the left or right depending on who is in power. So there's no quick database check.

However, in practice, if you're a citizen, it's pretty easy to prove it. You'll be carrying forms of state-issued ID and you'll have a paper trail that makes it obvious. There's rarely much ambiguity over whether a particular person is here legally or illegally, as much as some might like to pretend there is.

The US doesn't keep biometric data on every citizen or lawful resident, and the government can trivially lie or make a mistake about whether they did the database/ID check. This isn't a hypothetical, many legal residents and some citizens have been swept up, and without due process they have no ability to say, "I'm a legal resident" or "I'm a US citizen". They can just be shipped off to an El Salvadoran prison camp where the president can claim, "oops, I can't get them back because they're not in our jurisdiction any more".