Can someone become allergic to cookies? No. To a specific ingredient found in some cookies, peanuts for example? Yes. And there are ingredients which are very common to those confections we collectively call "cookies," such as gluten found in most flour.
So, can one become physically allergic to ALL perfume? Not likely.
To a specific ingredient in perfumes? Maybe.
And is there an ingredient in perfumes as pervasive as gluten in cookies? Yes, those would be water and SD40 alcohol. Fortunately, an allergy to water is unheard of. And an alergy to SD40 is also virtually unheard of. And since SD40 alcohol evaporates within a minute of applying a perfume, this would be a contact allergy affecting the wearer, not someone smelling that person even a few minutes after application.
And, while both water and SD40 are common in perfume, just as there are gluten-free cookies if you look for them, there are also of course, perfumes which cotain no alcohol or water.
So, a physical allergy to perfume in general is virtually impossible.
Some people confuse allergies to plants with perfume allergies since many perfume notes are plant-inspired, floral notes, spices, herbs, resins, oils, etc. But most inhalation plant allergies are to pollen. While plants often use scents to lure pollenating insects, the scent is not usually the pollen but a separate chemical or chemicals. When the fragrance chemicals are extracted from plants, the extraction process is so through that no pollen will remain in the extract. Also, many perfumes don't use the actual plant-extracts to obtain plant-inspired scent notes in perfumes. So, if a person is allergic to rose pollen, for example, they will most-likely not be physically allergic to the scent of roses and, even if they are, they will not be physically allergic to all perfumes with a rose note.
So, the conclusion is that given the diversity of perfumes, a general physical allergy to perfumes in general is unlikely.