Dealing with scratched floor tile
If it was easy to do, then the tile would become scratched up merely by walking on it. Your feet scuffing the floor with a little dirt there would cause scratches. Tile is used for a purpose - it has a tough, hard surface.

Next, you must consider if the tile color goes all the way through, or if it is simply a surface glaze. If the color goes all the way through, then you could erode some of the surface and still see the proper color. Often however, tile will be a different color underneath.

Best is to replace the tile. This is not as difficult as it seems, IF you have a replacement tile. Always keep spare tiles for just this purpose when a floor is done. (Also carpet fragments, etc.) Surprisingly, it is often possible to find matching tile in tile stores, even for older floors.

Replacing the tile requires chiseling out the grout (or using a grout removal tool) then a cold chisel will help to break up and remove the bad tile. (An air powered chisel is great here.) Clean off any adhesive still on the underlayment. Then carefully butter the new tile, put in place, and regrout with a grout that will match the color of your old grout.