I don't know how, but the bird dogs I have had could tell the difference. They would point pheasant and chukar after having been trained on nothing but quail. They would also retrieve doves, but not point them. Never had one that would point song birds.
My dog pointed dove a few times, when she was younger. Now she just retrieves them. I think they learn what you're hunting or not, and what's to be pointed or not, with experience. We just yell at the stupid sparrows, but we shoot at the quail and pheasants. Perhaps yours learned very early on!
So why are game birds game? I suspect it's based on the birds - which taste good, which are good to hunt, and which reproduce enough to hunt sustainably. Over time, they became "game". Dove are songbirds in some places, game birds in others. To me, while they do "sing", they definitely qualify as game birds.
