In our family, Kraft Singles is always called “plastic cheese” - both for the wrapping & the way it melts.
Philadelphia, born and raised… from “SouPhilly”.
We said American cheese if we wanted to give our parents agita.
American cheese at a deli meant the white square cheese, but many families would say “New York Square” at the deli. That was intended to mean American cheese and it was very white… and… duh… square. To them… unique it was. Every other deli item, it seemed, was round, and it was also what they were familiar with: Round meats and cheeses.
Not sure I’m helping here, but the only time I’ve heard ‘square’ is when people were asking for “New York Square”.
My take on the origins: Since many Italian immigrants/people/family I knew funneled through New York, and since their cheese context was based on round varieties from Italy, I never thought them calling for “New York Square” was odd. That was what they knew as American Cheese.
Hope this helps. I grew up with Italian immigrants in a little Italian neighborhood in Philly. Father Pasquale once sent a friend and me to the local Italian deli to get “New York Square” cheese. We ordered American cheese when we got there. He gave us a dollar for running the errand. My friend never split it with me. 
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Grew up in the late '60s and '70s in South Dakota. Mom never bought the Kraft singles, it was always Velveeta which we just called ‘cheese’. For awhile we did get ‘commod’ (commodity) or ‘government’ cheese which we also just called ‘cheese’.
Yeah, I learned that the hard way. I had a ‘cheese steak’ in The Bourse Food Court by Independence Hall and such. It was horrible. I wish I would have at least got one from the food truck that was out front, it may have been more authentic. It sure smelled better.
the way I look at it, American cheese is meant to be an ingredient or topping, while many of the classic cheeses are meant to be eaten as is.