What is the 2-inch layer of masonry under my bathroom tile?
This floor has been floated, a very common and traditional method for preparing surfaces for tile. To "float" is to apply a cement and sand mix similar to concrete but without rocks, gravel or coarse sand. A 2" thick float is not uncommon at all.

It is often applied using a "dry pack" method wherein only enough water is mixed to ensure proper set but wet enough to pack, shape, feather, and blend. 

Do not use pre-mixed bagged concrete mix. If you do not want to mix your own, use pre-bagged mortar mix and "dry pack" the area. I guess you could then use a thin layer of floor leveling compound to flatten, smooth, and blend the damaged areas and seams.