The way I see it, there is a political issue and a personal practice issue (if you will). The political issue will be a so-called turf war where the clinician may be pressured to claim one affiliation (APA vs. NBCC, for example) in order that the organization has a united purpose among their members. The personal practice issue would be more of an issue of, "What role are you playing from moment-to-moment?" When do you decide that you are working under your psychologist license vs. the counselor license, etc., whether it be private practice or a position you were hired for? There are slight differences in all of those fields when it comes to standards/scope of practice, codes of ethics, representation of one's credentials, etc. It seems like, in clinical practice, there is no real need for anything other than the most advanced and flexible license/affiliation/credential.
However, Dr. P.'s example of teaching is an interesting one. You would think that they would be happy with a psychologist as an instructor and supervisor, because most masters-level training allows that. I have seen the subtle preference for mentorship by professionals in one's own distinct line of training and that makes sense--if you're training to be a counselor (or, especially, an MFT or SW), a psychologist might not have the experience that you are seeking--or, they just may have it. It's similar with LCSW supervision for counselors, etc. However, we must also consider that having a variety of training and supervision contributes to a colorful learning experience.
Just thinking out loud...I have to admit that I was a little tickled as I read and realized that it was possible to have multiple licenses, because my training will allow for that someday. I think the academic component is the biggest advantage there.