Some songs have to make perfect sense (those with a direct message that the writer wants to be sure will not be misinterpreted), but certainly not all songs do.

Wait a sec... maybe we need a definition of "sense". Most of Lou and Peter Berryman's humorous songs, for instance, sound like nonsense songs because of the clever phrasing and use of language in the lyrics, but they make perfect sense literally. So, how are we using the term "sense"?