Robert Nagle of
Idiotprogrammer posts this interesting quote from Umberto Eco on Casablanca:

Casablanca is not just one film. It is many films, an
anthology. Made haphazardly, it probably made itself, if not actually
against the will of its authors and actors, then at least beyond their
control. And this is the reason it works, in spite of aesthetic
theories and theories of film making. For in it there unfolds with
almost telluric force the power of Narrative in its natural state,
without Art intervening to discipline it. And so we can accept it when
characters change mood, morality, and psychology from one moment to the
next, when conspirators cough to interrupt the conversation if a spy is
approaching, when whores weep at the sound of 'La Marseillaise.' When
all the archtypes burst in shamelessly, we reach Homeric depths. Two
cliches make us laugh. A hundred cliches move us. For we sense dimly
that the cliches are talking
among themselves, and
celebrating a reunion. Just as the height of pain may encounter sensual
pleasure, and the height of perversion border on mystical energy, so
too the height of banality allows us to catch a glimpse of the sublime.
Something has spoken in place of the director. If nothing else, it is a
phenomenon worthy of awe.”

I think this comes close to explaining the unclassifiable nature of the
film (which I touch on briefly
— a quality shared by many collective works of art, like Gothic
cathedrals, for example, which have a mysterious, chaotic unity even
though they were built by many hands over many generations following
only the most general and ever-shifting plans. 
I would suggest, though, that Ingrid Bergman is the central vault of
Casablanca — the stunning core phenomenon around which the other disparate elements cohere.