Vaughan says, "no":
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.
Two words: Sigmund Freud.
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