How can I fix this coating on the kitchen sink?
Looks like a repaint. (Those are popular in the real estate world, because they look good for a couple of weeks.)

Since it's failing, I'm going to go out on a limb and say that it was a crummy job and will probably continue to get worse. 

Three choices:


live with it
painstakingly strip and paint again. (Professional recommended... good ones use epoxy paints/equipment that aren't really DIY. A good job will last reasonably well, though not forever.)
new sink, though this often turns into a quagmire of new faucets, plumbing, counter, cabinets, flooring, paint, windows...