I know this post is rather old, but I thought I would add a few points as well.
Soapstone does not need to be sealed. Soapstone is, for all intents and purposes, chemically inert. This is the reason why laboratory benches historically used soapstone. Installers usually recommend a finish from #80 to #120, followed by an optional coat of mineral oil, ...
it looks like melamine or laminate.
Either way the stuff is heat resistant to well past the boiling temperature of water, those are not burn marks from a mug, so they are som kind of dye mark.
Use a strong bleach: both surface types are chemical resistant.
If that doesn't work use a gritty cleaning compound sparingly, the surface is only thin.
At first I thought porcelain tile but after looking closer I believe melamine is correct especially after looking at the slight cracks /fractures along the edge. I have a car polisher I use for quick countertop touch ups. The random orbital polisher has a large area and buffing compounds really work well at cleaning a surface like this or Formica. Make ...