When a person does not identify AS a non-human animal but instead has a strong "connection with" or feels a "relation with" an animal. "I identify WITH wolves." does not have the same meaning as "I identify AS a wolf." Being a Therian is more than "really liking" an animal. Liking a creature a lot does not equal an identity as that creature.
To use the term correctly, a person who feels close to deer would say, "I am deer-hearted."
There is a thread on AHWW from January 15, 1997 in which someone who claimed to "ABSOULUTLY ADORE WEREWOLVES AND WOLVES" asked if they were "wolfhearted".  The exact terms other-hearted and animal-hearted seem to have come into more common use around 2014 or 2015.
The equivalent term in the Otherkin community is other-hearted.