The shrimp burger keeps the burger's architecture but replaces its load-bearing assumption: the patty is shrimp, ground or chopped and bound into a cake, and it brings none of the fat that holds a beef patty together on the heat. That is the defining problem the sandwich is built around. A Lowcountry South Carolina format, it takes the bun, the cold acidic dressing, and the handheld logic of a burger and applies them to a fried seafood cake that has to be engineered to stay intact, because shrimp left to itself will not cohere into something a bun can carry.
The craft is in the bind and the fry. Shrimp is chopped or coarsely ground rather than pureed, so the cake still reads as shrimp and not paste, then held together with just enough binder to survive being formed and turned without packing it dense. It is fried hot so the outside sets into a crisp shell while the inside stays moist, which is the same crust-versus-center timing a fried fish sandwich solves and the same reason the cake is made thick enough to have a center worth protecting. The bun is soft and the dressing cold and sharp: a tartar-style sauce or hot sauce and a pickle supply the acid and crunch that a sweet, mild shellfish patty lacks, and the soft bun keeps from competing with a delicate cake the way a crusty roll would. The result is a seafood sandwich that eats with one hand, built on the coast where the shrimp comes off the boat.
The variations stay inside the bound-and-fried-seafood-cake idea and mostly change the catch or the coating. Some builds lean the bind toward a crab-cake density; others fry a lighter, lacier crust or swap the sauce toward a remoulade. It is one regional dialect of the American burger's non-beef branch, alongside the salmon and the fish builds that keep the architecture and change the center. Those deserve their own articles rather than being crowded in here.