Music is like a natural language in some respects and very much unlike one in others. Here are some suggested similarities and dissimilarities.
Music is like (a) language in that:
It can be described through a system of rules that operate on a limited vocabulary
It combines small building blocks into large components that are like words, phrases, sentences and text
It is recursively expressive
It has dual articulation in that smaller segments like scales are organized independently of large segments like movements
It has phraseology and idioms
It can cross-reference between compositions (texts)
It can communicate emotion both segmentally (sequences of notes) and suprasegmentally (expression, emphasis, etc.)
It has styles, genres and dialects
It can be acquired and learned
It is culturally conditioned
Music is NOT like (a) language in that:
It cannot be used to directly communicate propositional meaning
It has radically smaller set of building blocks and rules for their combination than language
It does not have internal instruments of disambiguation
It can only be universally acquired in the most rudimentary sense (i.e. everybody can hum a tune but very few people can play an instrument)
There is much a greater difference between receptive and productive competence
There is much greater variability in individuals' ability to produce music beyond the most trivial
Much more of the production process requires cooperation among individuals
It is much more limited in its dialogic potential (i.e. is most often used for a one way communication between few producers and more recipients or joint co-production of producer/recipients)
Similar lists could be constructed for other communicative systems where people talk about the 'language of X'.