Is it possible to have a fragrance allergy?

Yes : Nasal symptoms and even migraines can be triggered by exposure to strong odors, smoke or even changes in weather. Vasomotor rhinitis refers to sneezing, runny nose or stuffiness with these exposures. This reaction is distinct from usual allergies and responds less well to antihistamines. Astelin and atrovent spray or nasal cortisones like otc nasacort (triamcinolone) can help.

Yes and "sort-of": Topical contact allergies are not uncommon. Contents such as balsum of peru, cinnamic acid, and vanillin are common contact allergens and can cause contact dermatitis. For the airways, these, by far, more likely irritate. The airways in those with allergic rhinitis and asthma are inflamed/tender. Aerosolized substances with an irritant quality will afflict airways whether allergic or not.