Tile – Is a tablespoon of vinegar and a cloth adequate to clean tiles with compiled soap, dirt residue

In the to-do section of the care instructions you posted, it suggests "Use an all-purpose, non-oil based household or commercial cleaner." OR, it says, you can use vinegar/water.

I think you are over-analyzing. The normal floor mopping solutions like Pine-Sol or a Swiffer will work fine. Just don't use anything abrasive that will scratch the tile or an aggressive bathroom "tile/grout cleaner" that is meant to bleach and renew grout that is stained.

If you're concerned about a particular cleaner, you could always test a small section that is out of sight. You can also apply a "grout sealer" solution over the grout lines. This is tedious, but can help slow down staining of the grout.

This looks more like a cracked tile than a scratch (but it is difficult to say from the picture). Was there work done after the tiling that may have put stress on the wall? I must admit that it is unusal to see this on a wall tile. Replacing it is not rocket science and the preferred method (there is really no other way to fix this).