It’s not like there are people checking for immortals, I think it would be flagged by a dmv employee or something when they dont believe a clear 21 year old is actually 150. Let’s assume it’s current day im caught and not bring speculation on what the US is like in the year 2139 is like.
it’s not hard to commit identity theft in the US. people are just stupid about it.
I don’t think there’s a way to commit identity theft in this hypothetical that would work. Being immortal would mean it had to work in the long run, or at least a few decades until you can do it again. Someone will notice eventually. You can call it “people being stupid about it”, but a mistake will happen if you wait long enough.