Actually, shellfish and seafood are not synonyms.
seafood mass noun
Shellfish and sea fish, served as food
shellfish noun (plural same)
1. An aquatic shelled mollusc (e.g. an oyster or cockle) or crustacean (e.g. a crab or shrimp), especially one that is edible
1.1 [mass noun] Shellfish as food
As you can see, shellfish are a type of seafood, but there's a rather large class of seafood that isn't shellfish, namely fish. And then there are the squid-type creatures, which are neither fish nor shellfish1, but are seafood.
1 Well, actually, lots of people do consider octopus and squid to be shellfish, kind of by process of elimination: it's seafood, but it's clearly not a fish, therefore it must be shellfish.
(Where English tends to get sloppy is with freshwater fish: many a "seafood restaurant" will have types of fish on the menu that never saw any saltwater in their life.)