Dual diagnosis is an understandably confusing term. Diagnosis itself is confusing enough, and one might rationally conclude that dual diagnosis implies that two psychiatric diagnoses have been given. This is sort of true. While more than one psychiatric diagnosis can be given to a single individual (and commonly is), dual diagnosis refers to the presence of both a diagnosis related to substance abuse or addiction and one that is not (e.g. diagnoses more commonly thought of as "mental illness). A typical example is an individual with a diagnosis of bipolar disorder and a diagnosis of alcohol abuse.
Commonly (though not always) individuals with a dual diagnosis are seen as doubly broken. They're extra disordered. And therein one sees the limitations of an over-dependence on diagnosis. Diagnoses bring stigma, they narrowly define what an individual is struggling with, and consequently narrowly define their treatment. In the case of dual diagnosis, this is doubly so.
As psychotherapists, we would never ignore the emotional struggles that others define as "mental illness," nor would we ignore struggles with alcohol use and abuse. But the prevalence of diagnosis and the limitations therein are too constricting, alienating, and too narrowly define both the patient and the problem.