To the extent that an illness has some direct biological cause, like genetic inheritance, or being caused by excess blood levels of mercury, illness is not going to vary by culture.
On the other hand, both would we choose to respond to things and how we judge the sensibility of others is strongly influence by culture, so there's no way culture isn't creating different ideas of mental illness and even, in a sense, some different mental illnesses. I don't know about you, but when I look through the DSM, I often think (what a poorly organized lot of arbitrary bullshit). I know they must be getting at describing something real, but that doesn't mean the way they do it is particularly accurate, and the DSM and ICD can't even agree with each other. Culture is obviously at work in how these are drawn up.
Or consider this; I had psychology professor who had also worked doing in-patient care and as a prison therapist (what a life). He told me about what he called the Oprah Effect, which is that if a mental illness is given attention by some highly popular instrument of the media, the rates of people claiming to have it and even the rates of doctors diagnosing it spikes. That's some pretty flimsy authority right there.
I once read of a psychologist who, due to geography and word of mouth, acquired multiple Amish clients. He quickly found that it was challenge to attempt his profession as he was trained to do when dealing with people who lived in such a dramatically different culture. You can't necessarily suggest they do or not do all the things you'd tell someone to do or not do in typical American society.
We can't even agree on what a mental illness is here. Try to define mental illness itself. Go ahead.
So to say that mental illnesses are not discrete things like Polio strikes me as blatantly obvious.