I don’t know how it’s done these days. In the old says a driver wanting to pick up a car from impound had to go get a release from the the LAPD. They way around this is you are unlicensed is to sign a power of attorney for a friend to pick it up for you. In the case of illegals, chances are their pals have no valid ID either.
As far as insurance. If the owner has it any driver would be covered. There have been lawsuits over damage done by car thieves who have stolen cars owned by uninsured.