Bayes’ theorem says no such thing. Your friend is wrong, but please do not let this reflect badly on the Bayesian interpretation of probability in general—it has nothing to do with this.
And in fact a Bayesian approach, by being explicit about conditioning information, helps to show the “trick” behind the Monty Hall problem: when Hall opens a door, he reveals information about what’s behind the other two doors that the contestant didn’t have when he made his initial choice. The probabilities change in response to this new information.