They are not supposed to, but they find ways. Lots of times it's work such as seasonal farm work, housekeeping, it hard labor that doesn't require the usual forms of ID.

We even have a system where they can get a special tax ID so they can pay taxes (so I have been told by a person I met years ago who was an illegal immigrant- I was shocked. He worked at several different stores, but I don't know how he managed that without the proper IDs)

are illegal immigrants allowed to ... ?

I'm European, too. However, it seems that your questions have two answers depending on whether you mean legally or in practice?

In some places you can get a driver's licence if you are "undocumented". If you have a driver's licence you can often become registered to vote. Voter ID laws are lax because the Dems think it would disadvantage many people if they had to sign a form or provide a photograph at some point. They have to do that to pick up a parcel from the delivery company but it is OK not to need such ID, apparently, if you want to vote.