
Some of the visual elements in the levels were also meant to evoke a play. For instance, the platforms that seemed to be suspended by strings or bolted to the backdrop. Also, the way the levels all ended was supposed to be an "exit stage left" (which it would be from the perspective of Mario as an actor on stage.)
Nintendo does that a lot, actually. Look at the intro to Super Mario All-Stars or some of Nintend's online presentations. Those seem to be probably deliberate callbacks to Warner Bros cartoons that showed mortal enemies like Bugs Bunny and Elmer Fudd as colleagues and actors when off-camera("What's Up, Doc?") Mario was designed as an "actor" like Bugs or Mickey that could be plugged into any situation he was needed for.