They don't, though. Not for all valid immigration statuses. For example, while a friend of mine was here on a student visa, they simply had their visa, their drivers license (which not everyone would have), and their foreign passport.
I'd say it would actually be a fantastic idea to have some kind of immigration identification card for people who are here legally but are not peanent residents. Something simple that dumb law enforcement officers can accept as firm proof of legal status.
I actually don't think this sheriff is wrong when he says we need a simpler method of identifying whether someone is legally here. I likely disagree with him in everything else but I agree with him on this.