Yes, is the short answer. The global reference book on mental health, Diagnostic and Statistical Manual of Mental Disorders, lists the broad category of hoarding as a recognised psychiatric disorder. Within that broad category of hoarding falls animal hoarding. Therefore animal hoarding has been recognised, since 2013, as a mental illness
It seems to me that the experts have decided that all forms of hoarding, be it of inanimate objects or living creatures, can be categorised under the same or similar mental illness. Perhaps a better phrase is “psychiatric disorder”.
I believe that I have answered the question in the title and therefore have no more to say on this matter. However, feel free to add your thoughts in a comment.