They are often found in Civil War camp areas. The studs were punched through the leather much like a rivet. Cool find. You'll see many styles of those. Used as a harness stud
Saddle boss? A lot of these, with the same decoration, have been posted here and now I can't recall for sure what the final word was, except non-military horse related decoration. Someone posted concrete proof at one point, but I can't find the thread with it.