I mean no disrespect and certainly there are probably many physicians who do think psychiatry is fundamentally different from other medical specialties. However, I do believe that the roots of depression are very much like the roots of heart disease and diabetes. In heart disease, there are genetic makeups that predispose people to have heart disease. So too are there genetic predispositions to depression. There are theories of why heart disease occurs including the buildup of atherosclerotic plaques which may be affected by things such as cholesterol levels, obesity, or psychosocial stress. So too are there theories of why depression occurs, particularly that there is a paucity or excess of various neurotransmitters, or various deficiencies of brain function in certain areas of the brain. These factors may be affected by things such as early childhood experiences, exposure to sunlight, amount of exercise, or psychosocial stresses.
As far as your assertion that some mental illnesses may not be illnesses at all but perhaps the extreme ends of normal, I would agree that one could take that perspective. However, the definition of what makes something a "disease" is a social construct. Our society has typically defined things as diseases when they cause a significant impairment in one's functioning, or when they represent a significant threat to future functioning. For example, heart disease can kill you outright from a heart attack, or perhaps decrease your exercise tolerance. Psychiatry too has included impairment of functioning in the definitions of various mental illnesses. If a person is occasionally sad, but is able to function normally, we don't call that depression we call it normal variance of human emotion. If, however, they stop eating, have sleeping difficulties, are predominately "sad" most of the time, withdraw from social contact, and begin to think about killing themselves, we call it depression. I will grant you that it is not always easy to tell on cases which walk that fine line, but this is not a problem unique to psychiatry. What is the cutoff of a blood sugar to call a person a diabetic? It has changed many times over the years, to the point that they even now have points where they call people "pre-diabetics". When does someone have hypertension? Is it 140/100, 135/95, 120/80? It all depends on the patient and what study you prefer to take as correct.
Finally, yes psychiatry is often viewed as different from other medical specialties, but I believe this to be a function of two things. First off, it is a function of the stigma surrounding psychiatry and the belief that people with mental illness are "different", so therefore psychiatry and psychiatrists must be "different" as well. Secondly, it is a function of a philosophical premise which pervades our society in which people believe that mind and body are two separate things. I would contend (and I am CERTAINLY not the first) that they are, in fact, not separate, or at the very least they are intimately connected such that physical and mental illnesses must be seen as entities which ALWAYS affect one another. I truly believe that to view mental illness and psychiatry (and psychology for that matter) as fundamentally different from physical illness and medicine promotes the marginalization of the mentally ill and misses out on the opportunity to see ourselves as whole human beings with multiple body functions that all affect each other.