Recognized the material right away, looks like Corian also called solid surface polymer, stuff is brittle so don't drop your gun, it won't chatter but will bruise badly turning it white.
Works just like wood with files and sandpaper, you can go with increasingly fine grit moving on to wet sanding for a high shine, you can stop sanding wherever you like the finish and depending on how much you handle it, it will naturally polish up, you can get a matte finish using scotch-brite pads (red or white) green is too coarse.
I installed solid surface counter-tops for a living for many years and had some of this material sitting around for a long time but never thought about using it this way.