Why would my cousin be allergic to my cat when he was never allergic to cats in the past? he sneezes a lot.

Allergies change: Adults change their allergies on average every seven years. Just because you have been allergic to something does not mean that you always will be, and new allergies can happen.

Allergies: Allergies develop with repeated exposure. The more an individual who has a propensity to get allergies is exposed to an antigen, the more likely to develop allergy to that antigen. Your cousin is not allergic to your cat. He's become allergic to all cats, probably because of the repeated exposure to yours.