You can be a fake academic and build a career on plagiarism. However, stealing someone’s identity is dangerous, as academic fields tend to be quite small social networks, and someone is bound to notice. It’s also much more difficult to wiggle out of identity theft lawsuits than universities’ ethics rules.
It tells us something about psychiatry as a field. Imagine faking being a programmer or a physicist for 20 years: a trivial Fizz Buzz-like test would reveal the truth for the former, and the failure to reproduce research results—for the latter.
You should listen to the “dr death” podcast [1]. Season one was actually terrifying and is, in fact, what happens when an imposter pretends to be a surgeon.
Actually I say it's the other way around. In academia you're most likely to get caught. The communities are just to small for things to get unnoticed. You have many people reading everything related to the field, so people are likely to notice at some point. I'd say if the imposter would have gone for a PhD it would have come out at the latest.
Judging by some of the revelations in other business it seems not exceptional for people to add things to their cv's they've never done. And it's much less likely to ever come out, because people are less likely to check (and it might also be more difficult to check)
There have been several cases of people posing as doctors without any qualifications ("catch me if you can" is a movie about a famous case for example).
Fake being a surgeon and they’ll assign you to surgery, etc.