How can I clean this sink?

Is it possible that the finish has been eaten away? If that is the case, I don't think you can fix this. If it is just dirty, try closing the stopper and putting a folded paper towel over the area, pour vinegar and leave overnight. Scrub with an old toothbrush. This will clean just about anything.

Try CLR or Acetone, not nail polish remover, the kind you get at Home Depot or Lowes. There is a product called grout refresh that comes in many colors that may hide the blemish.

The finish has come off the drain assembly. The porcelain finish on the sink has worn off too. The drain assembly would need to be replaced. The sink can be painted with an epoxy tub and tile paint kit.

https://www.hometalk.com/search/posts?filter=paint%20sink

Here are step by step instructions on how to refinish your porcelain sink, the darkened areas look as if the surface has worn away. Also, the drain is in need of replacement.

https://www.instructables.com/Refinishing-an-Antique-Porcelain-Sink/

The drain assembly needs to be replaced. To try to make it look not so rusty you can close stopper and pour coke or Pepsi soda on stopper. Leave it there for 15 to 20 minutes. It will be much cleaner for awhile until u can replace the assembly. I do it occasionally and it stays for awhile. The sink is another story. Needs to be painted as stated in other answers