Is it possible to have winter allergies?

Allergies. Can occur year-round. In the winter it could be perennial allergens such as dust mites or animal danders.
Absolutely. Winter allergies can be due to perennial allergens like cat, dog, cockroaches, dust mites, mold/fungi. But you need to remember that cold air is a natural irritant...The colder it is the more your nose runs and swells. It is a protectant mechanism to protect those mucus membranes. See your allergist.
Winter allergies. The answer is yes. Winter allergies are usually indoor allergies - think about it - we spend a lot more time indoors in the winter. Mold can exist in heating ducts. Dust mites, Dog and cat danders.