I wondered that, but if you enlarge it you'll note it has a picture of him on it.
I think I can see where loops may have attached to work as a bridle rosette, but the rest of the back doesn't look like the ones we usually see here. I'm not very knowledgable about the varieties of these though.
I'm thinking it was a pin back, maybe celluloid type thing.
I find online that there seems to have been a lot of George Washington patriotic buttons made in 1932 for the bicentennial of his birth. -just a possibility
I can't find this button (or whatever it is) anywhere online, so it could be rare. I doubt if it's from 1932, though. I'm sure I could find another one somewhere online if it was that recent.
Looks very similar to many 1890s mourning pins I have had. They all have a celluloid or plastic covering over the photo, and have that metal braided type frame around the image. Is the image under glass or celluloid?