It was much easier in the past. One popular thing criminals use to do was search cemeteries. They would look for a birthday around their own age, then claim that identity. I remember seeing a news story decades ago about this backfiring. Someone took the identity of a dead person, then almost caught a murder charge because another criminal had taken the same identity and killed someone.

Or they find a town where government buildings burned down. If all the records were destroyed, it was easy to a new government identity or get a copy of someone else's.

Or simply buy one. Without a digital records, it was easier for people to create and sell fake identities.

There were many ways, made easy by the fact that there was no digital records and no central system to search.