Since it's actually etched into the metal, you'll need to buff the sink to a consistent finish. One method, assuming it's a stainless steel surface, is to use a mild abrasive with oxalic acid, such as Barkeepers Friend and clean and abrade the entire sink with a sponge or cloth. This will require a lot of rubbing.
If that does not produce a consistent finish, abrasive grits can be used, starting with a moderately coarse (low number grit) and finishing with one that gives an even matte appearance.
Do not use steel wool, which inevitably leaves embedded fragments that rust.