That's an interesting historical and semantic question.
So there may not be any such thing as a "wild pigeon". Pigeons were domesticated so long and widely that all modern pigeons in the wild may be the feral descendants of ancient domesticated pigeons. But we still view modern pigeons as just a thing that exists out in the world.
If street dogs have been a self-sustaining population for long enough does that change their classification? What if the population of street dogs is older than many of the cities they inhabit? What if the population is thousands, ten thousand years old?