Actually, even under the "correct" interpretation of the host's actions in the Monty Hall problem (where he knows where the car is and will never open that door

Actually, even under the "correct" interpretation of the host's actions in the Monty Hall problem (where he knows where the car is and will never open that door and the participant knows this), there's still an analogy, an even closer one, to this ambiguity. Just as here, if you ask the question in a certain way, the choice of the person, when they have the choice, becomes important (which gender to announce in case of GB/BG), in Monty Hall, if you ask the question in one common way of asking it, the choice of the host becomes important. The host can choose which door to open if they're both goats. If, in that case, the host will pick randomly, the probability of a win by switching remains 2/3. But suppose you picked door 1, and the host will always prefer door 3 when he can, for whatever reason, and you know this. Then given the information that he opened 2 or 3, the probability to win is 100%/50%, respectively. So if you ask the Monty Hall question this way: "I picked a door and the host opened another, what's my probability of winning now?", to get 2/3 the question should include the information on the host's random selection between 2/3 when the car is in 1. Admittedly that's a bit pedantic, but there you go.