Theriotype is the most common word for the animal(s) which is a therian's identity. (Therioside is also acceptable.) The term came into use in 1999. Some believe that a theriotype develops as part of their mentality much in the say way that their personality develops.
An example is "I have a whale theriotype." It is also acceptable to say "I am a whale Therian."
The term was developed as a replacement for the words Wereside and Phenotype. Some therians believe they have more than one theriotype, and are called polytherians, but this is somewhat uncommon, and often looked upon with heavy scrutiny from the community.
Mythical animals are usually considered to be Otherkin, but this has been disputed, especially in the case of dragons and gryphons, which are mythical beasts and also experience more instinctual and animal-like behavior during shifts.
Therian Timeline, Words and Concepts, http://www.theriantimeline.com/therianthropy/words_and_concepts
Terms and Definitions, Sonne, hosted on Project Shift. http://project-shift.net/terms-definitions