Whether sentimentals want to admit it or not animals are things and should not be treated like people. They are not family members nor do they have human emotions. People who dote on their animals make me sick, and it's these people who always seem to put their snooky ooky cutesy wootsies above other humans who may be inconvenienced or allergic. I truly care for my dog and would be crushed if something happened to him, but I don't think I'd make others have to sit by him on the plane. He'd be sedated and put somewhere else if possible. Actaully, at 16 he's probably too old to fly anyway. My only concern would be how the disgrutled luggage handlers would treat him.
I could not and would not sit next to a cat on a plane. I'm severely allergic to cats and I agree with whoever said it's my responsibility to be prepared, but since when do animals belong in a seat? So in answer to the OP, it is not reasonable.
Then again, I hate babies and small children and think they should be in stowage as well. :D I've been drinking for hours so i reserve the right to edit this once I sober up and stop being an ........

On reflection, I think I would be willing to sit next to a pet if the owner offered me appropriate sedation. :D

while I agree that pet allergies could be a concern...
 

what about MY allergies to certain perfumes, etc, that some people insist on bathing themselves in.

Cats I can take.

Perfumes make ME ill.

so where are the limits?

I'm more than willing to share the doggie downers that weiner dog gets when I fly with him...I'll notify you when I'm taking him on a future flight. :)