Therianthropy is today commonly known as a concept that encompasses those who are animals. In the past, and to this day however, there are few who claim to be mythical Therians, like a dragon therian for example. These people usually are referred to as Theriomythic respectively. By definition though, Therians are most expressed as being an animal that exists or once was known to have existed.
This phenomenon is important to each person, as the identity of a Therian is one that is integral to the individual, completely involuntary and unchosen, and something most believe they were born as. It is something ingrained in their nature as a person, whether the individual believes it due to spiritual means such as reincarnation, or rather a psychological reason that occurred in their brain's development. The identity is in no way physical, as Therians all know they are in human bodies. This doesn't change the fact that they are animals. Its not a relation, or having a favourite animal, it is being that animal.
The people who make up Therianthropy today, use the modern term "Therian" to identify themselves as such, but previously they used to call themselves "weres". The lingo obviously coming from lore like werewolves, the more modern term "Therianthrope" which was shortened to Therian, rather combines Greek terms that translates as "Animal Person".
These "animal-people" or "human animals" can still go about their daily lives, pretending to be human on the outside. Yet internally, they know and feel their animal natures.