There's some room for interpretation

He entered illegally, but also didn't have a choice... But he's also stateless, which is very bad and very messy

But then he was adopted by citizens, which grants citizenship... Assuming you go through the process. Which varies by iteration of Superman, but generally I think they just said he was their biological child

He's also an abandoned baby, which means he could get birthright citizenship by virtue of being found on American soil... We just kind of assume random abandoned babies were born here, because what other option is there? Making people stateless is an international human rights violation

But ultimately, he's not human. He doesn't get human rights, just like how non-human persons don't have rights (even if they have certain protections). It doesn't matter if an orangutan puts on a tie and gets a 9-5 job, even if they could speak and showed the ability to integrate into normal society... At this time, non-human persons are not legally recognized by the US

So... His situation is significantly more tenuous than some of the people getting deported right now. But he probably would have clean paperwork, because the state doesn't know the details of his origin and his adoptive father was smart enough to hide it from the state