Etymology of the noun-form of gamy / gamey [closed]

I was watching Futurama and there was a scene where Fry tasted a steak and said it tasted gamy (or gamey). Now I've looked up on several dictionaries and they all state that in the context of meat it describes a strong flavor or odor of game. However when I lookup the aforementioned word game and it's plethora of definitions there isn't a single definition that relates to any form of meat or even food for that matter. What does this game exactly mean?

This probably goes without saying, but it hasn't been directly addressed by any of the existing answers.

The word game, which originally and primarily means roughly a form of amusement, evolved to mean meat by way of hunting as a form of amusement.

OED definition 15 walks the line between the original meaning and the meaning referring to hunted meat.

15. Originally: †entertainment derived from the chase (obs.). In later use: the sport of hunting and shooting (or otherwise catching or killing) animals, birds, etc., as a countryside pursuit.

From there it took on the meaning referred to here, which developed the adjectival form gamey.

The flesh of a game animal or game bird used as food.

This form is first attested in 1658.

Gamy is an alternative spellings of gamey :

also gamy, 1844....Meaning "tasting or smelling strongly" is from 1863.

From game:

"the sport of hunting, fishing, hawking, or fowling" (c. 1300), thus "wild animals caught for sport" (c. 1300).

(Etymonline)