How to Repair Laminate Countertop Wear Near Sink
Find and examine an exposed edge on your laminate somewhere -- most easy to find is where the front edge abuts with a wood trim strip (in custom countertop designs) or where it wraps around the front of the countertop (more common with the manufactured type with integrated backsplash). You'll be able to see the dark color of the laminate substrate and also its thickness.

If the sink had completely worn through the laminate then you'd see not only particle board, but also a cross-section of the laminate subtrate. It would look similar to that edge examined earlier -- it would be kind of like looking at the layers in a sedimentary rock formation. But I don't see any depth or cut-away layers in your photo; instead it looks like surface discoloration caused by water deposits, food/dirt/dust, or even a little mold. (Molds can exhibit a rainbow of colors.)

Try detailed cleaning around the edge of the sink. Rub it with a cloth or a green abrasive/scouring pad. Scrub with a toothbrush or scrape gently with a razor blade, etc. (Your laminate looks like it has a strong surface texture to it so scraping may not work well.) You could try letting it soak a bit with a mild acid like vinegar before doing the mechanical cleaning.