Can people be allergic to each other?

Yes, people [ myself included ] can be allergic to other peoples natural skin oil and skin dander. Itchy burning rash develops on contact. Proved this many times with people who have freshly washed skin, no soap residue and again the allergic reaction occurred with contact. Proof that people can be allergic to other people.

It is also possible that one is allergic to something on the other person. For instance, perhaps one person has a pet and the other is allergic to that pet. Or the person uses a perfume, hair product, soap, or something on their body that causes a reaction. Also, fabric softeners or certain laundry detergents can cause allergic reactions. Consider all of that first.