Splitting, or breaking, is the separation of sauce, cheese, or other emulsion. As a milk product, cheese is a mixture of water, oil, and protein (and some sugars, fungus, coloring agents, details vary). Heat causes those elements to “split” and is the reason you can’t make a cheese sauce without some kind of emulsifier.
Premium American cheese, labeled “pasteurized process American cheese”, is mostly traditional cheese by weight (usually cheddar, often with Colby or others mixed in) with salt, color, emulsifier, citric acid, and up to 5% added dairy fat. That’s all the same stuff traditional cheese has except for the emulsifier (commonly sodium citrate or phosphate) which keeps it from separating as it melts.
Also all cheese is a “processed food” before anyone gets riled up about the terminology.