They're both connected in this instance. A person with a personality disorder is really on a continuum of insight in which schizophrenia is the most extreme example and represents a complete dissociative break with self-awareness.

I'm not claiming that the kind of dissociation a person experiences when psychotic can be treated by therapy. Pharmacology is essential, and actually I do think after awhile therapy can be helpful (I have family members who have had psychotic episodes too).

The lack of insight in both cases is a product of dissociation. In a personality disorder this is a "mild" chronic state (compared to psychosis) which is amenable to very long term (years+) therapy, unlike psychosis. It is why BPD people with quite severe dissociation or who have episode of severe dissociation report mildly schizoprenic-like problems with their identity (eg. thinking that they are multiple people, lost time, etc.).

I was assuming his BPD friend had this background state, which would not preclude treatment. Treatment would, in part, be to deal with that.