Yes, there is such a thing as having an allergic type reaction to temperatures. This is defined as a physical urticaria. Other physical urticarias include Solar, Aquagenic, Pressure, Vibration and Exercise.
Most reactions are pseudo-allergic. By definition, an allergy involves inhaling or consuming an allergen. Physical urticarias have no known allergen. Despite the terminology and medical definitions, systemic reactions can be life threatening.
Cold has an arbitrary definition based on an individual feeling. For a person with a cold urticaria, cold can be defined as any temperature cooler than their own body temperature.
You do not have to be cold to have a reaction to the cold; contact with cold can trigger a reaction.
You can have an allergic type reaction to both cold and heat simultaneously.
Most reactions considered anaphylactic are really anaphylactoid by definition.
Moving to a warmer climate as a treatment for Cold Urticaria is a myth. Warmer climates present their own issues for those with Cold Urticaria.