A couple of years ago I corrected a problem like this on porcelain tile.  It seemed as if the grout had somehow darkened the tiles.  It looked almost exactly like your picture.  I tried about six different ways to clean that stuff off the tiles. And finally Dr Fred came up with the solution:  Combine a surfactant (Tide for cleaning clothes) with an abrasive (I believed we used a honing powder) .  That will make a paste and we ran that underneath a stripping pad on a swing machine.  That combination did the trick for us; The dark lines disappeared.