My daughter has severe eczema, which causes similar problems on different areas of her hands/arms/legs/neck. It does seem that she gets worse on her hands after she pets the cats, but she does have a mild allergy and takes oral meds for that. The main thing is to keep your skin moisturized, especially during the dry days of winter!
After you pet your cat (and several other times during the day) rinse your hands in comfortably warm water, blot dry, leaving slightly moist and apply a mositurizing lotion. then rub it, rub it, rub it until it is mostly absorbed. Dab away any excess, if you like (or rub it on your legs as I do).
My daughter does well with cocoa butter and coconut butter which can be found at Wal-Mart for a very low price (a couple of bucks), or any drug store or discount/department store. Any good (preferably thicker) moisture cream should work.
If you have any question about it being the cat, you can experiment by petting with gloves on.
Just a thought...but I doubt that it is really the problem. Just keep your hands moisturized. If you haven't tried it, you can get cotton "moisturizing" gloves near the manicure area of most department stores, drug stores, etc...rinse/wash your hands, blot dry, put on a thick moisturizing cream and then the gloves.
Wear them overnight and you should see an improvement after a few days, if not the first day. I got mine at Wal-Mart, but I have seen them at Ulta, Target, Walgreens and other drug stores.
Good luck with it! Try to remember that winter is hard on the skin and moisturizers are a valuable resource!