It's not terribly difficult to buy a fake SS # and green card, at least based on my experience hiring people in one of my previous jobs, lol. You'd be amazed how many folks had papers that looked legit, but popped in our system as a duplicate SSN (only because we already had someone working under that SSN. If we hadn't, it wouldn't have flagged and we would have thought it was good). All it takes is some cash and knowing the guy to go to. And in a large immigrant/illegal community, someone probably knows and will share. There's a big support structure.
So your MC may not know anyone initially, but depending on where they ended up there'd still be folks from their originating country that would be willing to help them navigate.
Another person in my life, who shall remain nameless, came to the country with their parents on a travel visa. They stayed for almost 20 years before legalizing, paid taxes, owned property, held good jobs, had 401ks, etc. The person attended school, got work, drove a car--the whole nine yards. And believe me, there's no travel visa that's good for 20 years. Their parents never did go through the naturalization process, but my friend finally did when they hit adulthood. There was surprisingly little issue with it, given that they'd been in the country illegally for 16 years.