Um, a mental illness is defined by being dysfunctional to the patient (and doing things that are odd enough that society throws rocks at them doesn't count). So if the patient is spending her rent money on furry comics, or is consuming furry media to the neglect of food and sleep, you might have an argument that it's a mental illness. (And then, in the 2010s, the psychiatric community has been having to consider that exposure to toxic circumstances: overmonitoring at work environments, bad bosses, not earning a living wage, excess rent, may be factors that drive dysfunction externally, figuring more largely in mental illness than internal factors like heredity. But that is bleeding edge still.)

But just a fanatic obsession of cute furry anthros, even if it is extreme, is not a mental illness, in exactly the same way that a man who is sexually and romantically attracted to other men is not a mental illness. Or if we want to get Victorian about it, exactly the same way a woman who refuses to accept her limited place in society is not a mental illness.

So I can assure you from here, the way mental illnesses have been defined since the 1990s, being a furry is not a mental illness.