Therianthropy is the deep personal spiritual path (In my opinion), that you have connection to a certain animal (Which may or may not be extinct).
Therianthropes may have a "Spiritual connection" that they believe that they possess one or several animal spirits either that are sharing a body with you or may even just be your soul in general. Most likely if you have many animals sharing a body with you, you wont have a clumped up together soul full of several animal you connect to, but most likely you will just have many animal spirits sharing a body within you.
Some Therianthropes have the mentality of a animal that wasn't thrown away during reincarnation and they have certain animal personality traits. Even if this had happened this dose not mean that this is the only way to have a mentality of an animal, because you can have one even if you share or don't have and animal spirit within you. This mentality can even happen if your past life was an animal an that animal soul that you had didn't reincarnate into a human one after death.
There are many ways that a person can be an Therianthrope but you don't have to be one to practice this belief, in my personal opinion.