Your brass find's width, and its "straight" (not tapered) body, and two rivet attachments prove it is a civil war era US Springfield rifle's sling length-adjuster hook. The hook works similarly to a waistbelt adjuster hook, fitting into one of the holes in the sling for adjusting the sling's length. The photo below shows one on a modernday reproduction Springfield rifle sling.
After-posting edit:
MudHut's answer is correct. He beat me to posting because I had to hunt for the photo showing the sling adjuster hook on the sling. ;-)