Is that some type of grit applied to one side? If so, I'm guessing it's a nail file.
It might be bone or ivory or celluloid. Here's a good site to help you identify the material. distinguish bone, ivory and celluloid
To me it looks like a removed tile, the lines on the back look to have been applied with a toothed trowel? Bakelite has that look. Is there a chip on one edge? HH
BK