Everything in life is biochemical. Every feeling you have is biochemical. Every sensation, every memory, every problem. But this is sometimes exactly as meaningful as calling it "atomic", which it also is. Depression is no more strictly biochemical than it is strictly atomic.
You can adjust the aggregate neurochemistry of the brain all you want, but people still do not instantaneously get long-term behavioral or affective change.
Depression certainly can be viewed biochemically, but to do that is to rob it of its fundamental roots in our human experience, to ignore how it is so often caused by life events either big or small.
And we know that cognitive-behavioral therapies and a host of options that do not involve altering the chemistry of the brain directly are tremendously effective at treating depression.
In my mind, if we want to characterize a disorder as biochemical or neurological at its base, it needs to be the result of aggregate-level damage or long-term imbalance rooted in injury, heritability or some other factor. Bipolar disorder, for example, could be termed to be biochemical, because it does seem to be based on some kind of failure of regulation in the brain. But depression is both a symptom and a diagnosis, so we have to be very careful with how we classify it.