I found this the other day. I thought suspender clip thingy but it's very thin, delicate and bends easily. It not magnetic and it's measurements are about 1 and 1/2 inches long and 3/4 inch wide. Back is the same as the front. Google-image search yielded nada.
Thought you'd like to see the Boston Garter (patented in 1895) illustration, posted below... its shape is similar, but a little bit different. Note, at that time, garters weren't only worn by women.