game fish vs fish

game fish and fish both are nouns.

Word	Noun	Adjective	Verb	Adverb
game fish	Yes	No	No	No
fish	Yes	No	Yes	No

As nouns, fish is a hypernym of game fish; that is, fish is a word with a broader meaning than game fish:

game fish: any fish providing sport for the angler
fish: any of various mostly cold-blooded aquatic vertebrates usually having scales and breathing through gills

game fish (noun)	fish (noun)
any fish providing sport for the angler	any of various mostly cold-blooded aquatic vertebrates usually having scales and breathing through gills
	the flesh of fish used as food

game fish (verb)	fish (verb)
	catch or try to catch fish or shellfish
	seek indirectly