That is both technically true and materially false. Allergies are mediated by proteins but sometimes the problem is a natural protein altered by chemical reaction with an environmental non protein. Mercury allergies are this way. Smoke allergies could be.
Do a search on it. People do have allergic like reactions but not because of an allergy to smoke. It really is interesting. The article also stated that smoke CAN aggravate an allergy to other things like pollen.