Is it possible to be allergic to a potato?

Yes. Yes, you can be allergic to any protein. All living things have specific proteins, including a potato. So, you can be allergic to potatoes. It's important to distinguish, however, between an allergy and an intolerance - different symptoms, different causes, and different treatments.
Yes. It can cause itch, hives, swelling, and anaphylaxis. I have patients whose hands swell and itch when they peel raw potatoes, or eating raw ones cause mouth itch. These patients have tested positive. To get tested, take a raw potato to your allergist.