I would try using an automotive rubbing compound. Scratches are gone and the color is not affected.
First try buffing out small scratches with Bar Keeper's Friend Powder. I have seen this work wonders on white ceramic dishes.
The only thing I can think of to cover something more serious would be appliance paint (you can buy tiny bottles at the hardware store) but the shade may not match.
If you know the brand name of the sink, you can call the manufacturer for a bottle of scratch cover that is usually applied by a small brush (like nail polish). If not, some folks will mix artist’s oil paints until they match the color of the sink and then use that. With either way, you will want to thoroughly clean the sink, dry it thoroughly too, and lightly sand only the scratch to help the new scratch cover “cling” to the old scratch. If you use artist’s paints, be sure to note what amounts were used to match for any future use.
FYI: sinks and tubs can also be reglazed by a professional such as Bath Fitter. This is an expensive option. After factory deglazing products can also bd found in some high end paint stores and online. Follow the directions carefully and have super ventilation if you do this yourself. I did a tub once, and it came out great and lasted over 15 years, saving a ton of money. Good luck!