We need something in the pic for size reference but I’m on board with Tom. Strap goes through one end,the hoop on the other end gets a spring clip of some kind...
That is an old suspender clip. Yours has a patent date, which is very faint, but will read : PAT'D JUL 10, 88 or such. Probably from the 1880s to 1900 era
Here is one I found a while back.
I have a match to your buckle, but the tab is broken off of mine. I just dug mine up 3 weeks ago in South Central Iowa on a farm that’s been in the family since the early 1800s. (Now if I could only figure out how to upload photos!)
The above hyperlink will take you to the patent with schematics I found trying to match mine up.