Certainly looks like slate. Take a little screwdriver or nail file and, in an inconspicuous place, see if it scratches. Slate will....porcelean will not.

You need to find a good neutral PH cleaner.

Agree, it looks like a basic multi-color slate.

I'd guess it's a natural stone mainly due to the amount of lippage in the installation. A porcelain look-a-like could have the colors and cleft, but it'd likely be better installed due to manufactured tiles being more even in thickness from one tile to the next.

Ok, thanks! Do you think Home Depot has a good neutral pH cleaner for it?

Yup, that's slate. I would go to a tile store instead. They typically carry a cleaners for both tile and stone and the salesmen know much better than the clerks at Home Depot about which product to use for what.