I suspect that what you are seeing in martial arts movies is more of a cultural artifact than anything: A representation of what that culture considers valuable, as opposed to a representation of an absolute association. In capoeira music is played for (strongly) cultural reasons as a way of setting the style and the energy of the game, but not because it specifically makes you better at the art.
I will say that it is valuable to not become too narrowly focused. Don't just be a martial artist: go play an instrument, learn how to paint, learn poetry, learn a language, learn chess or go, etc. These can help you develop as a martial artist on the Way side of things and help you get more in touch with yourself, but should be taken as incidental the details of the practices: there are aspects in everything you do that may give you inspiration or your improve you as a well-rounded martial artist. In short: The specific choices don't matter, just that you do them. That you keep your mind engaged and active and seek to better yourself.
The details–which instrument, whether it is music or something else–is something that I would argue is largely secondary, though certainly some are going to have more direct application than others (specifically taiko and dance, which have already been mentioned). What does matter is that you enjoy it, even when it gets hard, and that you want to do it.