Regrouting, what to do with screw holes
I'd be inclined to either: fit another soap holder; remove the offending tiles and replace; fill the holes with grout, making sure it does more than plug the surface.

With idea 1, put silicone in the holes first, then the screw plug (deeper than the tile, please!) and screw holder on, with a touch of silicone to finish.

With idea 2, there's always a chance you can't get tiles to match, but if yiu can, it's the best option.

With idea 3, putting enough grout to seal the hole won't be difficult, but there will always be two blemishes on the tiles that won't go away.