Doesn't look like one to me. If the stub had a hole or threads, then I would say yes but from what I can see of your picture. I haven't seen anything like this on horse tack and I have owned horses along time. As far as a roofing rivet, looks to big and domed shaped for that. Could be a rivet that that use in steel boats though or anything metal. Thats my guess anyways.
It looks too well made and decorative to be a rivet ... I still think it is a part of a pull knob. I did a reconstruction of what I am guessing it is. Even though it isn't perfect, I think it gives the general idea.