The panino con cappon magro is an argument the bread mostly loses, and that tension is the whole interest of it. Cappon magro is the most elaborate construction in the Genoese repertoire: a tall, mounded salad of layered poached fish and shellfish, boiled and dressed vegetables, hardtack soaked in vinegar, and a thick green sauce of parsley, capers, anchovy, pine nuts, and oil, built up on a plate into something closer to architecture than to lunch. Folding that into a roll means compressing a dish designed to be tall and composed into something that can be held in one hand, and the sandwich is defined by what survives that compression and what is given up.
The craft is reduction, not assembly. The full dish cannot go between bread, so the panino keeps its load-bearing parts and discards the rest: a few pieces of the poached fish or shellfish, a restrained spoon of the green salsa doing the work the dressing and the layering did on the plate, perhaps one boiled vegetable for texture, and the soaked hardtack logic transferred to the bread itself, which is left sturdy and lightly moistened so it stands in for the dampened base. The portion has to be controlled hard, because the green sauce is intense and a generous fill turns the thing into a wet collapse before the second bite. The result is honest about being a compression: it reads as the idea of cappon magro, the sea and the sharp herb sauce held together, rather than the full ceremonial mound.
The variations are mostly about how much of the original is kept: the spare version that is little more than fish and the green sauce on good bread, the fuller build that smuggles in a layer of vegetable, the relatives in the Genoese cold-table tradition such as the stuffed cima. The full plated cappon magro is its own production and a different thing entirely, eaten with a fork and no bread at all. Each is a different Ligurian dish given a handle, and each deserves its own article rather than being crowded in here.