In the 1970s it was much easier to take on a new identity than it is today. Assume the name of someone who died young and didn’t create much of a paper trail, and won’t be using it, Perhaps someone that didn’t get a social security number when they were born (it was much more common then). Get a copy of their birth certificate, apply for a social security card.
I knew someone that did this, and not the guy in Catch Me If You Can.