Can I use stainless steel scrub to clean regular dishes?
You can but you probably shouldn't. Stainless steel scrubs are quite coarse, rough and with sharp edges. They are  particularly abrasive, suited for scrubbing the pan's metal surface and removing the outer oxidized or grimy layers and achieve a "shiny finish".

Used on ceramics I imagine they will scratch the glossy surface quite easily; with continued use it will eventually lead a faded finish and loss of gloss.

It will largely depend on the hardness of the specific ceramic type. I would assume average quality ceramic dishes, but even for a finer quality porcelain over a longer term it will probably eventually wear out the vitreous coating, and on extreme cases even reach the more crumbly inner substrate.

On a more expensive/high quality porcelain service I imagine you would want to use something softer anyway, to prevent premature wear, particularly if it is some sort of antique, heirloom, has any sentimental or monetary value.

Decorated items with paint or enamels will be particularly sensitive to abrasion, since even continued use in a dish washer will lead to faded colors over time, especially on over-glaze decorations.