It's rather arbitrary, actually. In Italy they regularly hunt a bunch of birds that we'd call songbirds.
For years mourning doves were considered songbirds in many states (including here in Montana) and were illegal to hunt. But that's changing.
A century or so ago almost all "shorebirds" were considered gamebirds in the U.S., including some that rarely went anywhere near a shore, such as curlews. Now about the only shorebird you can hunt is common snipe.
Mountain lions were considered varmints in just about every state half a century ago. Now they're game animals.
