Concrete slab was the foundation, original tile installation had mortar applied directly to concrete slab. Removal of tile with a pry-bar easily separated the tile/mortar from the concrete slab. Why was is it so easy?
Concrete, mortar or similar materials are not designed to stick or bond to old surfaces. You will not get any satisfactory results if you simply add new mortar to old. It just doesn't work. Using a modified thinset mortar would be the preferred method for this type of installation.
Same thing for grout. New grout will not stick to old grout. You need to use a bonding agent such as Grout Bond for repairs and re-grouting projects. Even then, you really need to make sure the area is completely free of dirt, grease and grime.
Make sure you take the needed time to prepare for any repairs grout/mortar related. As with most projects, the more time spent in preparation, the more professional your results will be!