This is not a question which is asked a lot and neither is it answered a lot, or at all usefully, but theoretically it is possible for a domestic cat to be allergic to another domestic cat. However, I don’t think is possible for a domestic cat to be allergic to the primary domestic cat allergen which causes an allergic reaction in people, namely the Fel d1 protein.
But that does not mean that a domestic cat can’t carry an allergen in their fur which can cause an allergic reaction in another cat. For example, a domestic cat may have gone outside and picked up pollen or an airborne allergen on their fur, bring the allergen inside and interact with a companion cat inside the home which causes an allergic reaction in the companion cat.
But, as mentioned, there is nothing to suggest that domestic cats are allergic to the Fel d1 protein which causes about 10% of people to itch and have watery eyes etc.. However, a study tells us that 10 cat allergens have been identified in studies of extracts from fur, saliva, serum and urine. The most important of these is the one I’ve mentioned, namely Fel d1, which is called a secretoglobin and a thermostable protein found in the saliva, anal glands, sebaceous glands, skin and fur of cats. It is believed that the main production site of this allergen is not saliva but the sebaceous glands.
There is nothing in the books and the studies that I have read which tells me that one domestic cat can be allergic to another domestic cat because of Fel d1. Google is completely confused by the question in the title or similar questions such as “Can cats be allergic to cats”. It simply cannot find anything on the possibility of cats being allergic to cats which indicates that there is nothing on the subject of note.
So if you are concerned about your cat being allergic to something and suffering from allergy I’d look somewhere else other than another cat. I would welcome comments on this subject please. However please make the comment as scientific as possible.
Here are some articles on cats suffering from allergies: