Yes, allergies are common in cats and dogs.
They arise by similar mechanisms and can cause similar symptoms, though skin eruptions are more common in cats and dogs than sneezing or digestive distress. Treatments may be similar, too.
In an allergy, the animal’s immune system is hypersensitive to some substances it encounters, called allergens. For example, it is very common for pets to be hypersensitive to certain proteins in the saliva of fleas.
Pets, like people, can be allergic to substances that are airborne and inhaled, like pollen, mold, and dust; to substances that they come into contact with, like wool or flea collars; and to foods, like corn, wheat, soy, beef, or dairy products. The allergic tendency can be inherited or develop after repeated exposure.
It would be very unlikely for a cat to become allergic to its owner, since humans do not ordinarily lick themselves clean the way cats do, spreading some allergy-causing protein in their saliva onto fur and dander that is then blown about in the air.
Treating a pet’s allergy may involve protecting the animal from re-exposure to the allergen, desensitization therapy, shampoos and rinses to remove an irritant from the skin, and the use of antihistamines and steroids.