According to the "size scale" object in the photo, which is marked 1CM (presumably, 1 centimeter), the circular portion of the unidentified object is somewhere between .4-in and .5-inch wide. That's too small to be a musical instrument's button/key-top. So I am in agreement with Indian Steve... it seems to be "the back of a rivet type button" -- also known as a tack button. It is the tack component, which during the button-manufacturing process gets pushed very tightly into a tube on the back of the button's front. See the photos below.