I looked into this when cat and I were stranded and trying to get home. The airline I talked to charged $100 for pets, but emotional support animals are free. My doc and I talked about it, but we were both kind if squicked by the ethics so we did not do it.
The reality was, I am his emotional support animal :). My cat has diabetes, and the stress of traveling without me (another option under discussion) would have been very hard on his health.
We ended up finding another way home, which was stressful but kept us together.
I used to think the emotional support animal things was BS, TBH, but then I learned how important it can be, for example for veterans with PTSD. So now I think it is only sometimes BS :B