Domestication is not an animal trait
I've noticed that a lot of "inexperienced" therianthropes who most of the time are the ones questioning confuse human traits with animal traits, as well as the usage of animals.
The urge to be on a leash, wear a collar are not dog traits or any other non-human animal traits. It is an important thing to remember when questioning, keeping out human "similarities" and seeing which are more animal. However, sometimes they can be a sign of domestication for canines and a few felines, as they've been domesticated for the longest of time and have changed a lot.
There is a higher amount of horse therianthropes than in 2017 and early 2018 when I only met a couple of them, none of them were riders. Children are getting access to phones/social media and there are the so called horse girls, too. They have seen riding, almost all of us have seen it and know it, but it is really not a horse trait. A horse didn't have the first tought that it needs to be ridden and should find something to jump over. Jumping, dressage (with exceptions, as classical dressage is based on natural movements), being ridden are not horse traits. Horses prefer to be with their herd outside and playing, rather to be together with us and be ridden, unless they are messed up by humans.
As therianthropy and our identities are natural, a jumping or being on a leash trait is not an animal trait and not part of the nature.