Once you have started playback of a Playlist, it is copied to Playing Now, and is no longer a Playlist. So no, there is no way to display or see the playlist that you are currently playing... because you are playing Playing Now, not a specific Playlist.
Think about it. You could add several Playlists, and some individual Tracks, to Playing Now. MC would need to keep track of where each file was added from, because MC works at the file level, not at the Playlist level.
What you can do though is add the "Playlists" column to the Playing Now View, and that will show all Playlists that the Track you are listening to belongs to. If it only belongs to one Playlist, then you would know which Playlist you are playing. If the Track belongs to several Playlists, then you have several Playlists that you could be playing.
If someone knows better, because I've never really looked into this, speak up. I don't mind being wrong.