The US is in the weird position of having a class of people (undocumented immigrants) who are often provisionally allowed to live there, known to their state in some capacity, and yet unable to receive some government documents that a permanent resident or citizen would be entitled to.

Europe doesn't really have that status. Either you're known to the government and can receive documents from it, or you're a criminal in hiding, avoiding any and all government offices.

No it's not like that in "europe". Plenty of people have been in the limbo state for years in portugal for example until the new government started expediting processes. I had several refugee friends which were in this situation and had local jobs and some forms of id but not others. Like having social security and a tax number but no official ID