The fish finger sandwich is a frying problem with bread attached. Breaded fish fingers are cooked until the crumb is crisp and the fish inside is just set, then laid on soft buttered white bread with a cold sauce, tartare or ketchup, and not much else. The whole build is arranged to defend the crumb: a soft bread that does not compete, a sauce that insulates rather than soaks, and the shortest possible trip from pan to plate so the coating is still crisp on the first bite.
The craft is contrast and restraint. The fingers bring all the texture, so the bread is chosen to be yielding and the sauce to be cold and sharp, a deliberate counter to a hot, salty, crisp filling. Sliced bread folds better than a roll here because it presses flat against the fingers and holds them in a row rather than letting them slide. Mushy peas, when they appear, are structural as much as flavour, a soft bed that stops the fingers shifting.
The variations come from the chip shop. The fish butty uses battered fish instead of crumbed; the scampi butty swaps in breaded scampi; the saveloy and the spam fritter put a fried sausage or a battered slice of luncheon meat through the same logic. Each deserves its own article rather than being crowded in here.