It's about how mental illnesses are less biologically based than we tend to think they are. Not about how other people have different biology.
I wouldn't say less biologically-based. The overwhelming evidence is that the majority of mental disorders have an organic basis - some kind of physical or chemical change in the brain.
What differs across cultures is how those changes manifest, are recognized, and understood by various populations. Koro is a particularly well-documented case, but the underlying problem appears to be an anxiety disorder - it's symptoms just aren't recognized in Western culture because that's not one of the ways anxiety disorders tend to manifest here. In all our great wisdom, we then classify it either as irrelevant (less so now than ever before), or as a different type of disorder.
The problem that the article refers to is that Western understanding of mental illness is being forced upon other cultures. In some cases, this may be a good thing - mental illnesses are not always treated with compassion, and people suffering from them are often shunned, imprisoned, or killed. Western society has made a reasonable attempt to curb those responses as a result of scientific understanding. The problem is when we export our recognized symptoms of disorder to another culture, overwriting theirs. Not only then does the disorder itself lose it's cultural understanding, but it may lose the cultural treatment and even increase the severity of the symptoms (as in the anorexia case).
The problem, as usual, is getting the biology and psychology of mental illness to fit together. The biology of mental illness is quite similar across the board (near as we can tell with what cross-cultural and genetic studies that have been done, at any rate). It's how the symptoms manifest and are understood by collective society that differs, and arguably calls for different methods of understanding and treatment by clinicians. Funny thing about mental illness - unlike many diseases, treating the symptoms while virtually ignoring their cause can effectively relapse or even cure the illness - a rare case where mind over body actually works.