Can someone tell me what's the function of the collar?

well for an omegaverse it would stop the omega from getting an unwanted pairing but here it's more like a comfort thing. The dom has comfort in knowing that others know that they "own" that sub while the sub loves it because it means the dom cares about them and they get to carry that symbol of love around which makes them happy due to their instincts.

well instincts/ actual feelings...