Modern resin composite sinks are incredibly tough, and will not usually be damaged by ordinary chemical cleaners - the risk is actually greater where wastes are concerned, so care is needed.
Oddly enough, black sinks are the most difficult of all when it comes to maintenance - they tend to show every trace of limescale, whereas other colours don't. Whatever you do, don't use bleaching agents - tempting as it might be.
You can safely try cleaners like Cif if the vinegar method hasn't worked, and at the same time gently scrape the limescale away with a plastic scouring pad. Keep at it, and when you get the sink looking better, introduce a drying regime - wipe the sink clean of all water after each use, it's the only way.
I gave the sink a spray and rubbed it last night, it came up very glossy like when it was new, and so far it is still shiny like when new without drying with the matt lime film, and the water forms into drops as on a waxed car. It did that when it was new.