Bell21... what’s the difference between a service and emotional support animal? Or just a plain old pet flying with their owner? Assuming they are all trained and given the opportunity to go to the toilet before travel - donkey and peacock aside - they should be no different. Arguably a service animal/pet in a container under a seat is safer than a large unrestrained labrador in an emergency! As for allergies, you’d get those no matter the designation (service, emotional, or pet).
I'd say there's a large difference between a Service animal and an ESA - I assume that a Service animal will be well trained in supporting their owner and are probably less likely to over-react in unfamiliar situations. I'm not sure what sort of training an ESA would receive, and I don't know how it could be regulated properly to actually make a difference. I don't know what the distinction would be between a pet and an ESA, but I assume, from what others have said, that there would be some sort of training involved and assessment of the passenger's needs. Further, I'm not confident in pets being trained for flying - for instance, I have an anxious cavoodle and there's no way she could ever be trained to fly (I would become her ESA!).
Also re ESAs, it would be problematic to even certify people as needing an ESA --- I've worked in and around the mental health field, and the criteria for that would prove interesting and probably rubbery and not helpful in practice.
Fundamentally though, when we pay to board a plane in a confined space with a designated seat, and no option to move around, I'm not sure we should need to accommodate other people's pets or ESAs in that environment. It sounds like, as others have said, that it's ok most of the time, but it's unclear why that variable is needed in a plane. It's quite different to most other environments in which you might come across an ESA or pet.
If, however, airlines feel people need the options of animals on planes, it would need to be tightly managed - for instance, possibly depending on the size of the animal, that passenger should be eg required to sit up the back and purchase an extra seat, or possibly if another passenger is inconvenienced, this passenger should be entitled to request a different seat away from the ESA or pet etc
Also, what are the safety protocols in case of an emergency - leave your personal belongings and ESA's behind? What if the ESA gets in the way of emergency procedures?
It seems that animals on planes are much more common in Europe and US, but i suspect that Europe, like US, may need to tighten their guidelines considerably over the years to come --- like many things, it will probably take some really unfortunate incident related to pets or ESAs that provokes serious changes.
{I should add as a post-script that I like animals and I'm an easy traveller, so I probably would be quite unaffected by them relative to others, but i recognise that people have allergies, anxieties (whether about flying or animals), etc, and not to mention the invasive natures of smells etc particularly on longer flights, plus non-service animals could potentially become quite unruly in certain situations on planes}