Start with a gentle sponge and dish soap. ...
Wipe the sink down with baking soda. ...
Use vinegar to remove watermarks. ...
must use a non abrasive cleaner. I use white vinegar and dawn.
Use lemon juice and table salt to get rid of rust stains.
I use baking soda and either a Scrub Daddy or Magic Eraser.
baking soda & peroxide
Bon Ami cleanser will not scratch porcelain.
Is it possible that the top layer of porcelain has been worn off?
Just an afterthought, but there have been cautionary statements against using cleaning products with the ingredient "D-Limonene" on porcelain sinks, because the ingredient can wear through the porcelain finish, over time.
If you manage to erase the marks in your sink, consider coating the porcelain with a very good auto wax, several times a month, to preserve the finish.