Have I suddenly developed some horse allergy or is it allergy to just one species?

Maybe. Allergic reaction to horse typically involves eyes (itchy, red, teary), nose (itchy, stuffy, runny, sneezy) and can involve chest (cough, tightness, difficult breathing, wheezing). All horses are the same species (as are dogs, cats) and all breeds are equally likely to cause allergy symptoms. There is no breed of horse, cat, dog that is hypo-allergenic; that concept is a complete myth.