As identified by cognitive anthropologist Pascal Boyer, the single common feature of all religious is a preoccupation with unseen sentient beings, of which psychiatry says nothing.
In fact, mainstream psychiatry remains firmly materialist - usually re-explaining experiences that many people attribute to spirits, forces or unseen influences as biological dysfunction. So, in the most fundamental sense, the practice of psychiatry is typically contra-religious.
You could argue that this is 'replacing' religion through colonising the spiritual sphere of explanation, but this makes it no more a religion than physics or evolutionary biology.
Andrew Sullivan responds with two words:
A better example from the psychiatric model might be Jacques Lacan, with his "Big Other." At any rate, I think that Pascal Boyer's definition of religion is deeply flawed, as is evinced by Vaughn's and Sullivan's differing interpretations as to how it applies. Because "preoccupation with an unseen being" is a statement of content; religion, on the other hand, is a definition of 'form.'
I don't think religion is necessarily preoccupied with an unseen being. I happen to think that a religion is whenever you take a philosophical standpoint (it could be Boyer's "unseen person" but doesn't have to be), entrench it in a hierarchy with a codified dogma. That is what separates "spirituality" from "religion"; spirituality being a branch of philosophy (beliefs about the structure of the universe--content).
Psychiatry clearly doesn't qualify as a religion under that structure; there are (arguably) no universal "dogmas" and there certainly isn't a heirarchy, per se (the American Association of Psychiatrists or what have you don't really qualify). It is, like science, an empirical branch of philosophy (beliefs about the structure of the universe).