If you select the correct door, switching will yield an empty door.
If you select an empty door, switching will yield a prize.
You have a 2/3 chance of selecting an empty door, thus, 2/3 of the time, switching will grant you the prize.
Now suppose that Monty will always open the empty door as advertised, but he will also always prefer to open a door with the smaller number, if he has a choice of two empty doors to open. If that is how Monty always acts, then given that he's opened door number 3, your probability to win by switching is 100%, and not the usual 2/3. If he's opened door number 2, then your probability to win by switching is 1/2, not the usual 2/3.
If Monty will always choose which of the two empty door to open by random, then the probability to win by switching is indeed 2/3.
There is a way to phrase the problem differently, so that it becomes an unconditional probability problem, and then it doesn't matter how Monty chooses the empty door to open. One way to enforce this interpretation is to phrase it as follows: Suppose some player always chooses to switch. Over many attempts, what will be the approximate proportion of his wins? The answer will be 2/3. But that's not how the problem is usually phrased.