In a medical context, "I own the patient" or "patient ownership" signifies a doctor's responsibility and commitment to the patient's care, including making decisions, managing their health, and advocating for their well-being. Link
If said out of context, I don't think I have heard this set of words used together before. And I know it is not an expression, so I guess the medical context gives good meaning to what is being said. To own a patient is not something peculiar or special that anyone would want to claim. But then, the doctor can say that a patient is his and not necessarily own them.
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