What can I use to repair scratches on the edge of duraceramic tile?

I would find a permanent marker that matches one of the colors, like the gray, and just mark over the scratches.

How to Repair Holes or Cracks in Ceramic Tile

Patching unwanted holes in glazed tile like this can be a tricky proposition. Here's a solution that may not always be perfect, but will make a big improvement. Polyester resin or auto body filler, often used to repair car bodies. Mix up a small batch by adding some hardener and stirring it thoroughly. Using a putty knife, press the resin into the hole slightly overfilling it. Then just as the material begins to harden while it's somewhat rubbery, trim off the excess with a razor blade.

Next use 600 grit sandpaper to sand the filler smooth. Wetting the paper with water makes it work best. Now it's time to try and match the color of the tile. A porcelain touch up kit which is available at most home improvement centers comes with several different colors of paint. You can also use artists acrylic paints available from Michaels, Hobby Lobby, art supply stores, and online.

The trick here is to mix these different colors together until we come up with one that matches the tile. Apply the paint with a small brush. You can test it on a nearby tile. When you get it the way you want it, then just paint over the patch and let that dry. Give it one final sanding with that very fine sandpaper and you'll have, well, maybe not a perfect match, but something that's a heck of a lot better than what you started with.

You can paint your floors. You will just need a heavy sealer that is made for floors.

I've repaired ceramic tile "dings" like yours by painting over with nail polish. Once dry, you can safely clean the floor and it won't come off.