I found this object along with many other items dating back to the mid to late 1800s. It's about the size of a quarter and about 1/4 inches thick. It has a fancy flowery design around the edge and a mesh like material in the middle.
I have to agree with the Professor, likely a late-19th to early-20thc coat button.
Here's a similar example I found years ago at an early homestead, although my find is slightly earlier in date.
Victorian/Edwardian era perfumes were oily and stained clothing. These buttons had a piece of velvet in the center on which a lady could place a few drops of her perfume so she could smell nice without spoiling her clothing.