You obviously have no idea what you are talking about. Perhaps "Europe" to you only refers to some rich Scandinavian countries?
In Eastern Europe ID cards are frequently issued solely on the basis of witness statements, fraud is very common.
There are probably hundreds of thousands of "fraudulently" issued European passports in circulation. If you go walk near embassies in Chisinau, you'll be approached a plenty of people offering to help you procure one.
Even the UK has struggled with this https://www.bbc.com/news/uk-58876645
>And why would you live under someone else's identity? Isn't it identity theft?
Because you don't have your own, there are a plenty of Europeans who have never even had a birth certificate (or lost theirs).
Because yours comes with the wrong nationality, a semipermanent black mark that prevents you from seeking normal employment.