This is quite common and pretty harmless. The scratches you see don't go very deep, nor are they very wide. My All-Clad saute pan is nearing 10 years old and has a ton of micro-scratches on the interior. It still performs beautifully.
That said, the scratches can grab onto proteins and cause sticking. However, this is simple to prevent with both oil and proper pan preheating.
When a pan is preheated properly the metal expands, essentially closing all of the micro-scratches. This prevents the proteins from grabbing onto them and getting stuck. You obviously need oil/fat to assist with this as well.
To properly heat a pan to the appropriate temperature I suggest using the water drop method. If you put a cold pan on heat and drip a drop of water onto it, the water will sit there for several seconds then boil away. As the pan gets warmer this will happen more quickly, fizzling away in a second or so. Once the scratches start to close something weird happens.
First, the drop of water will break into a few mini drops which scoot around the pan as they evaporate. This is a sign that you are almost there. When the drop of water stays whole (mostly) and scoots around the pan like a mercury ball, this is the perfect temperature. I the water instantly vaporizes on contact, you've gone way too far and need to let the pan cool down. At this point you should add your oil/fat, swirl it around, and immediately add your food. (Make sure the mercury ball of water is gone before adding oil).
Also note that the mercury-ball phase is definitely too hot for unclarified butter, and may be too hot for some extra-virgin olive oils. They may instantly smoke upon adding.
Again, it's important to have your oil and ingredients in place (mise en place) before you start. It's quite easy to skyrocket past the mercury-ball phase if you have to open your oil, pour, and then season your ingredients.
"Stainless" steel is an exaggeration at best. It will stain. Chances are you have some level of mineralization in your water (hard water), and that's what's causing the discoloration.
If it really bothers you you can use Bar Keepers friend or something similar to polish it, but it's unlikely that anything that doesn't come off with normal washing will come off in your food.
Best Answer
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.