Due to the object's size (in relation to the coin shown in the photo), and having two spots on the back where a metal bar was broken off, it isn't a button, it is a horseharness rosette. Specifically, it is a US Army horsegear rosette from 1902 into the 1930s, showing the Great Seal of the United States. Here's a photo showing the backside of an intact one, and another photo showing one on a horseharness "headstall."