You can tell your cat is allergic to fish, or any ingredient, by the way, they react. In fact, cats can be allergic, sensitive, or intolerant to a number of ingredients, which includes not only fish and shellfish but also grains and additives. Typically, a cat with an allergy will exhibit a variety of issues, such as skin lesions or itchy skin, as well as digestive concerns. However, the best way to know for sure if your cat is allergic to a specific food is to consult with a vet. Sometimes, it helps to even have cat food for a sensitive stomach.