My take... Pitcher is responsible for the pitch thrown, regardless of who's calling the pitches. I don't mean that he has carte blanche to throw what he pleases, I mean that it's always his pitch. As such, pitcher has a right (and need) to participate in the process of what's called to some reasonable degree. If he shakes and gets the same pitch, then at least he voiced his concern... which might have to do with the pitch call, the location, the fact that he doesn't feel he "has" that pitch at the moment, etc. This gives pitch caller, catcher, and pitcher something to discuss later.. One or all participants might learn something for future situations in this way.
If pitcher is shaking repeatedly and this has been an issue such that the coach felt pulling the kid was called for... Then fine. But if it's some sort of "absolutely no shake" policy... Then the coach is wrong, IMO, and should go play xbox baseball instead of human baseball.
Bigger picture, I always like to see coaches teach their catchers to call games... all sorts of good reasons for this.