The eczema probably IS an allergic reaction, but whethere to the cat or something else, hard to say; however, dont for a moment think that a food allergy will only show up all over the body. My son is deathly allergic to milk (milk based formula included) and he had anaphylaxis that only have hives and visible swelling of the ears and face. He is allergic to rice cereal and he only got hives on his face. In fact, the only time he gets hives elsewhere is when something he is allergic to touches him somewhere other than the face. Anything ingested causes him hives on the face.
Before getting rid of your cat, visit an allergist and have him tested. Dont let them tell you that he is too young - anything that will cause hives is enough of an allergy to show up on a skin-***** test. My son was tested at 8 weeks and again at 5 mo, and both times he had positive results to those things that are problems. It has been an invaluable tool to helping keep my boy healthy!
You also need to keep benadryl on hand and know the dose - anytime he gets hives give it to him and also keep and eye out for wheezing, general face redness or swelling, or swelling of the lips and tongue. If you see any of this call 911 and keep whatever caused the reaction to take to the hospital and doctor.
Angel, et al
Angel, et al