Sorry to have to tell you, your brass hook find is not a "civil war sword hanger." It is the brass snap-hook from what is called a link-strap, which was used to connect the harnesses of two or more horses. The hole in it was for the small iron rivet which held the iron
Your find appears to be an exact match with the one in the photo attached below. It is from the civil war time-period, but not used only by the Military.
As you see in the photo, the hole in it was for the small iron rivet which held the flat iron "tongue" in place -- which is missing on your find.