How many different categories of fish allergy exist? Is it salt-water vs. Fresh-water fish?

No category. There isn't any categorization of allergies between fresh vs salt water fish. Allergy to one fish does not mean you are allergic to all, but may have higher risk of developing such. Shellfish does same thing, but the incidence of allergies to shellfish is higher. Symptoms vary from minor irritation to hives to anaphylactic reaction/shock. Be cautious and consult your doc... Good luck.
Fisn allergy. I am not aware of differences in toxicity of fish depending upon salt water or fresh water origen. The toxicity comes from the fish itself or the parasites they carry and the toxins they produce. Some have neurotoxins, some may cause asthma or even a serious urticaria or anafilaxis others just a rash. Some classification distinguish other seafood from fish.
Fish vs. shellfish. Within fish allergy, there really isn't a separate category. The parvalbumin protein in the fish muscle/meat is the usual allergen. The protein can vary in amounts and vary depending on processing. For more info, please see my blog at: http://www.Familyallergyasthmacare.Com/2013/11/fish-allergy/.