What are the sheets typically found in the back of cheap MDF cabinets made of?
The backs if low cost shelf units like that are made of several different materials. If the shelf came as a put-together kit in a box it most likely had a back that had a fold or two. These are usually made of cardboard with one side printed with a fake wood grain image on it. 

Other ones that came as a whole back piece will typically be a product made of pressed wood fiber (similar to a Masonite product or other pressed board). These will come with one side printed or colored to match the shelf if one side is exposed to view when the shelf is in use. In some cases where the the unit is more of a cabinet the back will just be a plain brown or tan color. 

The back shown in your picture, being white in color, looks to be a melamine covered hardboard. This is still a pressed wood fiber product but is harder and more similar to a tempered Masonite type product. If you do not care if the back is white or not you could replace the back with the tempered type hardboard. It will usually be a dark brown color, typically comes in a 4'x8' sheet and is easily cut with a circular power saw or a hand saw.