The fish finger and cheese sandwich is defined by the one thing the others avoid: heat applied after assembly. Add cheese to fish fingers and you have committed to melting it, and melting it changes the whole sandwich from a cold-sauced build to a hot one. The crumbed fish is the usual constant, a baton of white fish in a brittle orange coating, but the cheese forces a different problem. A slice or a grating of cheddar laid over the fingers and put under heat slumps, runs into the gaps between them, and sets the row into a single bound slab rather than separate batons. That fusion is what the sandwich is for. The cheese stops being a topping and becomes the mortar that holds the fingers together.
The craft is timing the melt against the crumb. The fingers have to be fully cooked and crisp before the cheese goes anywhere near them, because the coating cannot be re-crisped once the cheese has wept moisture across it. Cheddar is the usual choice for the way it flows when hot rather than splitting into oil and grain, so it pools around the fingers and seals to them instead of sliding off in a sheet. The bread is soft white that can take a brief grilling without going to leather, and butter to the edges still does its insulating job underneath, because the base is now fighting both fish oil and cheese fat. The sandwich is closed and grilled or toasted just long enough for the cheese to run, then eaten at once, while the inside is molten and the crumb has not yet fully given in to the warmth around it.
The variations are about what shares the melt. A smear of brown sauce or ketchup under the cheese cuts the combined richness with acid; sliced tomato or a few onions grilled in add a sharper note against the fat; the whole thing taken into a toastie press becomes a sealed, crushed version of the same idea. The cold-sauced fish finger sandwich and its tartare and ketchup readings are the relatives that refuse the melt entirely. Each deserves its own article rather than being crowded in here.