It looks like the documentation isn’t super clear on this point, sorry about that! You cannot share a playlist with a Group entity, only regular Shotgun users and Client users.
What the Client Review Site will do is remember the list of people you’ve invited previously. So if you were to Invite Client Users A, B, C, D, E, and F to view a playlist, the next time you went to share a new Playlist and began to type in “Clint User A”, you’ll get an auto-complete suggestion to either add that user, or the list (A-F) with whom you’ve shared previously.
Let me know if that helps explain the behavior, and we’ll work on making the documentation more clear.