In Hungary, it was extremely common. Up until the late medieval period, no one really gives a damn about small game (I know of exactly zero references to them). Once serf system was in place, there are records of nobles explicitly allowing their serfs to hunt small game, most common being rabbits and squirrels.

Things like cats, dogs, most birds and rodents weren't really eaten, unless the situation was really desperate. Birds have most exceptions, though, some birds were explicitly forbidden from being hunted because nobility wanted to hunt them (phaesant being the most prominent).

That said, such a meal would be somewhat rare, since you don't have the time to hunt that much when you're a farmer. Common folks still ate a lot more meat than pop culture would have you believe, but most of it came from livestock, most common being chickens, ducks, geese, rabbits and pigs, with the occasional cow or bull thrown in.