The American panini is defined less by what is in it than by the machine that closes on it. A ridged press applies heat from both faces and weight from above at the same time, and that combination is the whole sandwich: the bread is branded with grill lines and driven crisp on the outside while the filling is warmed through and compacted into a single bonded slab. The American version takes this method and detaches it from any fixed Italian filling, so the press, not the recipe, is the constant. Whatever goes between the bread, the rule is that it has to survive being flattened and heated without weeping out the sides.
The craft is in matching the filling to the press. The bread is a firm-crumbed loaf, ciabatta, focaccia, or a sturdy sliced bread, chosen because a soft sandwich loaf would compress to a wafer under the plate while a hard roll would never crisp evenly. A melting cheese is almost always present and is doing structural work as much as flavor work: as it softens it glues the layers together so the sandwich holds its shape when cut. Wet components are the hazard, so tomato is sliced thin and drained and sauces are spread sparingly toward the center, away from the edges where pressure forces liquid out. Heat is managed rather than maximized: too hot and the lines char before the center warms, too cool and the bread goes leathery instead of crisp. Built right, the press turns a cold stack into a hot, sealed object that eats in clean halves.
The American panini overlaps the griddled-cheese family, where the close relations differ by how heat and pressure are applied. Grilled cheese is the ungridded, unpressed baseline, melted slowly in a pan. The tuna melt runs a bound filling under the same logic; the patty melt puts a thin beef patty and griddled onions on rye; the bistro croque monsieur layers ham and a cheese sauce and finishes under a broiler. Each of those is its own sandwich and deserves its own article rather than being crowded in here.