Can an allergy to cats be deadly? Can someone have anaphylaxis?

With acute exposure. Someone allergic to cats who lives with cats will not likely have acute anaphylaxis but may have symptoms of chronic allergic rhinitis and asthma. When a patient with severe cat allergy who doesn't live with cats enters a home with a cat acute symptoms may occur. This may include throat closure (laryngospasm). I have never heard of this resulting in death. But it's not fun.
Not likely. Severe cat allergy may cause asthma which could lead to respiratory problems. Since avoidance is a appropriate step, patients usually remove themselves from exposure as well as using controller and rescue medications. I doubt that cat exposure would ever lead to symptoms of anaphylaxis- skin, cardiac, gastrointestinal, or circulatory.