The fish cake sandwich is built around a filling that is mostly potato, and that fact decides its whole character. A British fish cake is not a fritter of pure fish but a bound patty of mashed potato and flaked white fish, seasoned, shaped, coated in crumb or batter, and fried until the outside crisps and the inside heats through to a soft, starchy paste. Put that between bread and the sandwich is doing something none of the other fish sandwiches do: it is carb on carb, a potato-heavy patty inside a bread carrier. The cake holds together as a single unit rather than a row of fingers or a loose fillet, so the sandwich is a patty build, closer in engineering to a burger than to a fish finger sandwich, and it is constructed on those terms.
The craft is heat reaching the middle and the patty staying whole. A fish cake is dense and cool at the core before it is cooked, so it is fried slowly enough for the centre to come up to temperature without the crumb scorching, because a cold, claggy middle is the failure this sandwich is most prone to. The cake is sized to the bread so it sits as one even layer that the slices close over flat, and because the patty is rich and starchy the bread is kept soft and plain so it yields rather than adding more density to an already heavy mouthful. Butter to the edges does its usual insulating job against the patty's fried fat. A cold, sharp sauce is doing real work here rather than decorating: tartare or ketchup cuts a filling that is otherwise soft, mild, and uniform from edge to edge, and is the main thing keeping it from reading as one flat starchy note.
The variations are the rest of the chip-shop counter and the sauce question. A salmon or smoked-haddock cake swaps the white fish for a stronger one; the Thai-style fish cake brings chilli and lime to the same patty idea; tartare against ketchup is the usual sauce divide. The battered-fish butty and the crumbed fish finger sandwich are the relatives that drop the potato entirely. Each deserves its own article rather than being crowded in here.