Well, there is a huge HUGE variety in cheese quality in general. American cheese is no different.
US Government cheese is American cheese, although while classic American cheese is made from colby and jack, a wider variety can go into Government cheese. I cannot speak to the quality of it at all, having never had it to my knowledge. But at least it should be better than Kraft singles (which do still have a place, imho, just a small one. lol. Like... fine on a burger or whatever)
Anything single-wrapped in plastic is almost certainly "cheese food" or whatever.
In the deli, it will come in typically five or I think ten pound blocks - blocks, anyway. It can be sliced like any cheese.
To me, cheese is many different things. I enjoy your standard cheese types sold in most grocery stores, although I also enjoy cheese from a dedicated cheese shop on a wholly different level. I have a number of favorites that I will have a couple of ounces of along with some fancy crackers, good sweet gherkins, black olives, various things like that - or a good crusty bread or a nice challah or something.
So deli American cheese doesn't rise to THAT level, altough just tonight I had some great smoked gruyere along with Boar's Head yellow American and sweet gherkins for a snack, and the American held its own in that mix. heh.
I would say that I personally prefer a nice sharp cheddar - anything below medium is boring, so even a good quality mild cheddar is pretty boring to me. It's not a fair analogy, but I'd say the cheese food stuff is like a mild cheddar - fine but boring. Gimme a nice extra sharp and we're talking. Well, gimme a good deli American cheese and it's one I'll eat on its own. Lesser stuff is only okay if it's melty on some sandwich or something.
I hope that helps a bit. lol. I'd imagine your local deli should have some sort of American cheese, though I don't know that for sure. It's just that it's really common for sandwiches, so if they have a decent cheese selection, surely they will. lol. Board's Head makes a good one, Land'O'Lakes is pretty decent. I can't think of any others, but I have forgotten a common brand or two that should exist. heh.