On my Golf 6, the AUX input on my RNS 510 navigation system didn’t work, quite suddenly, after I started the car. Even a car restart didn’t do anything. Also, the “AUX” button was greyed out and just displayed “MDI”, which is the more general interface (your AUX cable is just hooked up to that, internally).
Turns out, this is most likely a software bug and I still have it from time to time: the AUX seems to deactivate itself, even if it clearly is activated. To fix that, go to the following menu (the names are just inferred from the german translations):
SETUP button, Media, then tap on “Enable AUX-In” do deactivate it, and tap it again to re-activate it. Music should start playing now.