A friend needed some cleaning money so I put her to cleaning the shower stall, which has on point blank and white alternating tiles. She used Scrubbing Bubbles, and I think unfortunately, a 3m abrasive cloth, like one uses to clean dishes in the kitchen. Viewing the black tiles from an angle after everything dried shows some of the black color (dye?) removed, but not all the way through to another color. Is there any way to restore the color to a uniform black without replacing each tile?

No. Glazed Porcelain can not be stained or painted successfully.