Absolutely not a sword-hanger... despite what some Ebay sellers say. It is indeed a part of horse-harness ("horse tack"). Specifically, it is called a "swedge style" gag runner... part of the harness for a buggy/wagon-pulling horse's head. You'll see the match-up for it (top row, third from left) in this scan from a very-early-1900s J.M. Eilers & Co. horse equipment catalog.