Those look like rust stains. Try wetting the area, then sprinkling it with Bar Keeper's Friend cleanser. Don't scrub it. Just let it sit for a few hours. Keep it wet.

You might also try soaking a cloth with white vinegar, and putting that over the stain. You might have to leave it there for a day or two.

I think these have come with use over the 20 years these countertops have been around. They are yellow discoloration marks in quite a few places.

Then it is probably time to replace them. No way to know if those are burn marks, rust stains or chemicals from personal care items.

I would try Bar Keeper's Friend or a stronger phosphoric acid cleaner (those used for removing hard water deposits.) If those don't do anything, then you can rule out rust.

Buy a new vanity top. That’s less than $100 for the size pictured for the same level of quality. https://www.lowes.com/pd/25-in-White-Cultured-Marble-Bathroom-Vanity-Top/4729017

live wire oak - it isn't a small vanity, it is a long one that features two sinks and has a custom fitted cabinet below it. It would have to be custom made which makes everything very expensive.

Personally, I would be tempted to get some acrylic paint and sponge it over one of the stains to see what happens. Or a bottle of fingernail polish in a close shade and dab it on. And, there are epoxy paints that can be used on cultured marble.

But really, a slab of granite and two drop-in sinks is the answer, although I totally understand your circumstances from your user name.

Time for a new vanity and top. There’s no point in putting new anything on top of old vanities. And that’s the mouse’s cookie. No point in putting a brand new top and vanity with the old mirror and hollywood lights. Or the tired old tile and cultured marble tub and shower. You can phase it in some, but it’s time for a complete redo. Better to do all the messy things at once.

If burn marks I don't know. Any sanding or abrasive would be last resort. I've had crazy luck removing rust marks in fabric (I won't go into why I had to do this frequently for a while) using lemon juice and salt, no scrubbing, just make a mixture and let sit, repeat if needed. Maybe commercial rust remover, maybe Mr clean sponges.

Try baking soda and steel wool. Maybe whitening toothpaste?

Miracle Method franchise or a custom cultured marble company will have a refinishing service. They can make it look new again. It's one of my favorite things about cultured marble. I've used these services for clients with excellent results in the past.