Although I agree that a different name for this type of depression could be useful in a medical context (as the root cause of post-partum depression is likely

Although I agree that a different name for this type of depression could be useful in a medical context (as the root cause of post-partum depression is likely to be different in the father vs the mother, so the distinction does matter), "post-partum" as a modifier can clearly apply regardless of who is going through the "partum". e.g, the father is experiencing depression after the birth of his child, therefore "post-partum depression". Edit: and as others have pointed out, a separate name is probably not needed as medically this is generally referred to as "post-partum paternal depression".