There is a deep scratch running up the middle of several granite tiles on my customer’s floor. is it possible to remove this scratch and blend the repaired area into the rest of the floor?

Anything is possible, but in this cause it is very difficult. If the scratch is deep, there is a possibility that you will have to grind very deep which will put a depression in the floor. If this is case it is better to replace the scratched tiles. If the scratch is light, it is possible you can remove the scratch and polish the granite to blend with the remaining floor. The exception would be with black granites. Black granite is very difficult to polish and until technology develops that makes it easier I would stay away from trying to blend a scratch in black granite.