Mental illnesses are kind of an odd subject. Animals do suffer from them: Here and here are some randomly selected articles. But one problem is deciding what exactly is abnormal behavior and whether or not it's a symptom of mental illness. Another problem is animals (in the wild, at least) with very obvious and severe mental illnesses probably just die. Both of these are outlined in the articles.

With humans it's very different. We define mental illness as "a diagnosis of a behavioral or mental pattern that can cause suffering or a poor ability to function in ordinary life."

We have a very good idea of what "ordinary life" is and it's clear to us when somebody is unable to function properly. We're also able to care for these people and observe what's happening to them (as in they're not just wandering off and dying without anybody noticing).

But even with humans it can be unclear what exactly constitutes a mental illness. Things like dementia, schizophrenia, psychosis, etc., are very debilitating and don't serve a practical purpose. Others aren't so obviously detrimental.

Modern society is a very new thing in terms of human evolution. We've designed our own social norms and not everybody fits into them. But does it really mean these people are cognitively abnormal?

One example I take issue with is ADHD. The societal norm (if we look at children) is to sit at school and do homework for long periods of time. Our distant ancestors didn't do this so I find it strange that we expect everyone to adhere to it. There are many arguments like this one that suggest it's a genetic trait with distinct advantages in certain environments.

You could probably make this argument for some types of anxiety, some personality disorders, etc.

And as /u/StumbleOn stated, some forms of depression (and probably anxiety) may be a direct result of individuals trying to conform to the societal norms we've invented.

Also, I didn't really cover this but people can be born with them or they can arise if the brain doesn't develop properly. This may be genetic or environmental but probably both.

tl;dr:

it's hard to tell if animals (in the wild, again) have a mental illness. If they're alive and functioning can we really call it a mental illness? If they're dead we can't really study them. We also can't tell what they're thinking.

With humans it's (somewhat) easy to tell if people are functioning properly. Some of these people do have clear mental defects but others may have functioned perfectly well without our modern societal norms.