I believe it is called a neck.
swan
Ring necked pheasant.
Feathers that are on the neck of a bird.
You are referring to the Wattle...It is a red flap of "skin" that hangs down under the beak.
My bird
ostrich (:
The bird with the biggest neck is a flamingo
The thing on a tui's neck is called a throat tuft or throat patch. This distinctive feature consists of a small cluster of white, curly feathers that contrasts with the bird's dark plumage. It plays a role in attracting mates and is a characteristic feature of this native New Zealand bird.
It could be a bird in the pheasant family. Ring-neck pheasants have primarily brown bodies with some mottling of colors with a blue to blue/green head. One variation of the ring-neck is called an Afghan which looks like the ring-neck but has no white ring on the neck.
The bird with a red head, white neck, and black body is a red-headed woodpecker.
To my knowledge there is no symbolic meaning of the action. It was an old practice to hang a game bird by its neck to "ripen" the meat, when the bird fell, the meat was ready to be cooked.
No bird has fur, birds have feathers.