The finish isn’t coming off, something else is transferring onto it. Metal pot rubbing, aluminum cooking sheet. Something. In my last house a couple belt buckles would leave marks like that (lot inside the sink obviously)

Inside of sink: metal from pans, cookie sheets, etc

On counter edge: belt buckles, metal button on jeans, etc.

Your first picture looks exactly like a starved miter joint that is filling with contamination, however, I can't imagine using a miter joint at a sink with curves on engineered stone. A solid surface top (Corian) wouldn't need or want a miter joint anywhere in material without movement.

I'm going with the metal mark theory. Try a little stone slurry or some Magic Eraser please.

As to a fix and prevention, a better edge detail with a more rounded edge and a nicer finish would be much more likely to resist metal marks and chipping. If the edge were reprofiled and refinished, the marks would be eliminated and hopefully the adhesive void, if there is one, eliminated.

Use some baking soda mixed with water and it will come right off.

Is there a scratch feeling that your fingers carch in it? i had that in an old house with caesarstone.

@H202, yes to the touch, it felt like it was a scratch in the surface. But I'm now thinking these marks have appeared in areas where there's a slight indentation in the surface. So it's very slightly lower because that's how it was fabricated, and likely this is why marks are happening in these areas.

(Not that it became slightly indented because it got scratched.)

Definitely going to pay attention to the fabrication of my edge profile when my countertop is installed!

@evelynsmom

If you use quartz in your new kitchen, I'd recommend getting Mrs. Meyers baking soda cleanser, which is essentially baking soda and water but in an easy to use form. It will get out most things from quartz. Turmeric stains, for one. It also removes any marks you might get from water rings etc. 

We have nail details on our counter swivel stools and let's just say the combination with kids in the mix means our counter ends up with metal marks constantly. Gets them right off.

Whatever you do don't use vinegar.

Thank you, @anj_p. Will use baking soda or Mrs. Meyers after this. Hubby is a big fan of using vinegar to shine pots and pans, etc. I just told him just now to be careful not to let vinegar spill (and remain) on the countertop.