The 'made with' is a dead giveaway. The double smoked aged cheddar I'm munching at the moment is not "made with' real cheese, it IS real cheese. And mighty tasty I might add.
Edit: Some fun facts.
British colonists made cheddar soon upon their arrival in America. By 1790, American cheddars were being exported back to England. The British referred to American cheddar as "American cheese", or "Yankee cheese", and post-Revolution Americans promoted this usage to distinguish their product from European cheese.
Even though the term “American cheese” has a legal definition in the United States as a type of pasteurized processed cheese
, products called "American cheese" are by no means identical.