The word stock originally meant a wooden post. Farmers would keep their stock animals behind a stock fence made of wooden posts. The word stock then became associated with anything you kept a stock of. Stock also became an adjective, so a stocky person is someone built like a tree trunk.
Why would Stock Doves be called stock, then?
It’s probably because they nest in holes in old tree trunks.
I like this photograph of a Stock Dove. They’re normally quite difficult to photograph, as they are very active birds, but luckily this one was perched stock-still.