Yes, it is possible for people that only has one eye get allergies. But, don't be optimistic, Usually, it will probably develop to two eyes, usually, 2-3 days. So, if you find one of your eyes get allergy, you should treat it immediately and avoid it spread to the other eye. It is serious. You shall visit an eye doctor and get prescribed medicine to help you cure your eye allergy.
Yes, it is quite possible. This is usually the first symptom of eyes allergy. You should go to see the doctor as soon as possible to get the treatment for the eyes. Or else, it may infect another eye because of the connected nerve systems in our body. If you treat it immediately, one allergic eye will get well fast. Wish you good luck.