I’ve had this same issue, and tried lots of things. Bar Keeper’s friend, that orange paste crap they sell, all of these are abrasive and scratch the surface of the ceramic, allowing the stains to get worse.

The answer is two-fold. To clean it, fill the sink with very hot water, like near boiling. Then, mix in some powdered oxi clean and let sit for 15ish minutes. You will be amazed when you drain the sink how clean it is.

The second step is to reseal the sink with a commerical product, I found several on Amazon. This will prevent it from happening again, but not as well as a when the sink was new.

Hope this helps you.