filling holes in bathroom tile
Leaving this as an answer with trepidation and the hope that a real tile expert will chime in.

My guess is that they're limestone or travertine. If you have spares, grind/smash/pulverize them to dust, mix that with a 2 part epoxy and fill 'er up. (Same for the broken chips at the door -- glue them solidly down with granite-safe adhesive and fill the cracks after the glue cures.) I don't have personal experience with this product, so not a recommendation (hopefully, someone else will chime in), but I'd be willing to try "Tenax Travertine Filler" (available on amazon). 

Hope this helps.