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.
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.
And that mistake gets more and more likely over time. An immortal would have had a much easier time 100 years ago. Even much more recently my first drivers license was paper where i was able to change my age with a pen (probably helped that it only had to fool bartenders). Now you leave a trail everywhere