Not according to EU law. Over here, Kraft Singles could only be described as "processed slices (made using a blend of cheese, vegetable oils and milk proteins)".
As long as consumers are clearly informed about what they are eating (via labeling) and the food is safe for humans to consume, constructing a legal definition of products like 'cheese', 'ice cream', etc. just seems like pandering to interest groups.
If it weren’t forbidden, this product would be sold in a package showing cows and cheese wheels, with “CHEESE” in huge letters on it, and, in a place people wouldn’t look at and in tiny low-contrast letters, a remark that this is processed protein made from diary.
Why eat something so intensely boring, when there is so much great cheese out there?
The answer to your second question is: Texture. The way process (not "processed") cheese melts is definitely different than similar cheeses. There's a gooeyness and meltability that's unique for applications like a grilled cheese or burger, and good process cheese is just cheese that's been melted with a few salts.
It turns out there are many different American cheeses:
Pasteurized Process Cheese - Various cheeses melted together with an emulsifying agent.
Pasteurized Process Cheese Food - Mostly cheese, but with a higher % of 'the other stuff'.
Pasteurized Process American Slices - Not actually cheese. A vegetable oil horrorshow.