When the Imagineers say "Everything in the parks has a story", I don't think they mean that it has to be a literal, complex, narrative. It can mean "story" in a simpler sense, simply defining what something is, and why it is the way it is.
So, the "story" of the Tiki Room could be that deep in the heart of Polynesia, there is a wonderous, enchanted room where birds sing songs. The room is so enchanted, that eventually the flowers and tikis join in the singing and celebrating. This creates such a rukus that "the gods are angered by all the celebrating", and cause a storm, ending the show.
Not as complex a story that would be needed to fill up a movie, but still a "story", from a certain point of view. I guess they could use the word "concept" here instead, if you prefer.