The prime requisite is not the form, but the intangible combination of animalistic and human qualities. Many people have specific furry characters that represent them. They are characters that they have created as representations of their furry personality. These attributes are the same as those of role-playing fans, who will come to identify with their various characters. Personal furries may also be favorite or totem animals, which that person simply likes.
Some people find that their personal furry is a representation of some aspect of their own psyche.
The division between their character and themselves exists only in the minds of psychologists. Other fans may have personal furries which are their favorite character to draw. They may often draw a particular furry of their own design, but experience an attachment no deeper than that. Fandoms are sometimes carefully organized, with membership rosters, newsletters, and such. Furry fandom is a specific subdivision of the broad Fantasy fandom.