Roughly 10% of the population are allergic to animals. This number is larger among asthmatics.
Cat allergies are more common than dog allergies. Of course there are allergies to other animals such as hamsters, rabbits and more.
The cause of allergy, as opposed to the mythos, is not the hair of the animal, but the animal's skin particles and different proteins in their urine and saliva.
Avoiding exposure to animals is the appropriate solution. If you cannot do this fully, then at least the movement of the animal should be limited, preventing it from entering the bedrooms.