I don't know much about bowls and dishware, but as a chemist I do know that emalox is a ethylene oxide polymer. It's probably very sensitive to any type of organic solvent, like acetone (nail polish remover) or toluene (paint thinner). If you've damaged the surface of the polymer, a very dilute solution of either might clean it up by removing only the topmost layer of polymer, or it might take off the polymer completely, depending on how thickly it's applied. It's kinda hard to say.

I would think the bowl would be undamaged by any sort of food-based cleaning agent. Vinegar isn't a strong organic solvent and as long as it's pretty dilute, it shouldn't harm the polymer. But it might not actually remove the discoloration. Another thing to try might be a bit of dilute lemon juice. Soaps can be a bit basic, depending on the brand and formula, so a mild food based acid might work.

Hope that helps a little. Someone else might have more experience with emalox in particular, rather than trying to extrapolate from the physical and chemical properties of the polymer.

I'd guess it was anodized aluminum with some kind of wax. Applying the right kind of wax with the same refractive index as aluminum(hyrdr)oxyde (the anodising layer) would give it a full shine again.