They do sometimes use scraps from other cheeses, probably as much as they can, but what really makes it processed is emulsifiers, which keep it from separating when melted, and additives that make it last longer.
I must admit that while I love a good slice of real cheese, processed cheese does have its place. For instance, because of the way it melts, it usually works very well on sandwiches like hamburgers.
However, when it comes to snacking, I prefer a nice Brie or perhaps some goat cheese. We usually have a nice array of cheeses come thanksgiving, which go well on crackers, and processed cheese never makes the cut.
The key when buying cheese, at least in the US, is to pay attention to whether it is labeled as cheese or cheese product, with the latter not containing more than 50% cheese. Some processed cheeses contain 100% cheese.
I always kept "American cheese" slices for my preschoolers because it was only the "cheese" they liked. All three of my kids were known, however, to occasionally get slices out of the refrigerator, unwrap a couple, and stick them on the refrigerator (like magnets). If you're food sticks on the refrigator chances are it isn't the best.