Is this hard floor mat in our closet made from asbestos?
It is difficult to tell, but it’s the right age. The EPA regulates the testing, abatement, and disposal of asbestos. The one thing I’ve learned over the years is that it needs to be friable to be harmful. This is not friable and you can easily remove it if you like. 

However, I’d be careful not to break it up and I’d wet it down before I tried to move it. That is to say, I’d spray it with water and wrap it in a large sheet in case it did break up. 

Also, you can buy test kits to determine if it is asbestos from your local hardware store. If it is asbestos, call your local sanitation department and dispose of it properly.