It depends on the airline and the situation, I'm sure, but in my experience, it's still the person with the allergy. Cabin pets require an extra fee, and since no actual service is provided for that fee, my presumption is that the fee is intended to cover increased liability and possible passenger reaccommodation for exactly this reason (notwithstanding the "because they can" factor).

I fly with my dogs in-cabin once a year, and one year a passenger made a big fuss (to the GA, not to me, and not directly in front of me) about the dog. The GA told her she could switch seats or take a different flight; I was not asked to do anything. (For the record, our dogs stay in their bags the entire time, and on most flights, even our row-mates don't know there's a dog on board.)