You might want to leave it as is because it actually looks like a normally patinaed copper sink with a "living" finish.

Unless you want to commit to shining and polishing it regularly, it will inevitably patina.

I just installed an apron front kitchen sink with a living patina and it looks pretty much like the one in your post in terms of patina. You can get rid of the patina with acidic stuff - e.g. in a kitchen sink lemon juice, tomato sauce, ketchup will lift of the patina but the patina will return in a short amount of time.

Do a search for copper sinks on houzz and you will see examples.

Your sink doesn't look as if it had the artificial dark finishes which some manufacturers used for their copper sinks - especially the ones with hammered bottoms.

Lemon juice and table salt. Pour, sprinkle, mix & scrub! Shiny like a new penny!

Tried lemon juice and salt and tomato... nothing worked.

I’ve used some of these to clean my copper pots -maybe one will work for you?

https://www.mr-maid.com/blog/5-easy-ways-clean-copper-cookware/

Some copper comes with a protective coating and it wouldn't surprise me if the sink did. It wears away unevenly and stains, etc. The only way to remove it is with a lacquer remover. Try one of these and see. First method won't work, of course, but try #2 and/or #3:

Removing lacquer from copper

eta, looking again, is this really copper? i've never seen copper faucets and the faucets and sink look like the same color/material, or is it just the photo?

Is it possible the sink is brass? or bronze?

Flitz metal cleaning polish, microfiber towel and elbow grease. Trust me.

Revere or Noxon cleaners have been know to work well.

The patina is why people purchase sinks with a living finish. Relax and enjoy it. Don't sweat the laugh lines.

#bringsophieback

https://www.gardenweb.com/discussions/5513253/bring-sophie-back#23104977 

https://help.houzz.com/s/profile-support?language=en_US 

I used to have unsealed copper countertops. They were a beautiful warm color, but every time I wiped them my sponge was green, and so they never felt clean. That might be a way to tell if your sink is copper or if it is sealed - I'm assuming the copper would constantly be oxidizing, and therefore would give off green if unsealed.

My countertops would eventually get tacky and I would use Barkeeper's Friend, plus elbow grease, to get them to smooth, shiny penny. But they would start to darken within the hour. I did enjoy how they always changed depending on what spilled on the counter, but I enjoy my new Quartz countertop even more because it feels clean!

I would also encourage you to learn to love the patina, especially if you have scrubbed to no avail, or just bite the bullet and replace the sink with something you can be comfortable with. =)