Our pets mean a lot to us. It's well-established that pet owners generally have higher morale and well-being than do those who do not have pets. Most of us love our animals like children and are concerned for their health just as we would be a child's. So when our pets seem depressed, unhappy, or stressed, we feel for them.
It's an extremely controversial subject, but there is a large contingent of people who believe that pets, like humans, can suffer from mental illness. Some are easier to spot than others, of course. A dog with attention deficit disorder (ADD) is, well, a normal dog because the problem is not really a doggie issue. A chronically depressed dog, however, is not generally normal. Neither is a dog who adversely reacts to certain normal stimuli due to past trauma.
Pet Psychology
A growing industry of "pet psychologists" and "veterinary behaviorists" is appearing. Like anything on the fringes of acceptance, there are plenty of quacks among the otherwise honest practitioners in this field. Some are striving very hard to gain acceptance and show some science behind their work.
The battle for acceptance is uphill, however, as science knows even less about animal brains than it does human ones - and we've barely scratched the surface on our own psyche. Like human psychology, much of the growing pet psychology research is based on observation and thus can be subjectively skewed or hard to replicate.
Mental Illnesses in Pets
The most common and well-accepted pet psychology issues are various forms (or manifestations) of anxiety, a few phobias, aggression, and handful of compulsive disorders. Most are somewhat similar on the surface to the human counterpart illness, but are both harder to "read" and more difficult to pin down because of the inability to communicate and far fewer case studies from which to draw information.
But the idea that pets can have at least some mental illnesses is gaining ground with pet owners. This is leading to more research funding and better study.