"This question can be extended ..." -- that's absolutely right. But of course, if you want to step all the way back -- that's the case for every phenomenon. Every time you flip a coin, it gets a little dented, and changes the likelihood of coming up heads. Every time you shoot a basket, your arms are a little more tired (or a little better rested) and your chance of the ball going in are just a little different.
As an applied statistician, an enormous part of your job is trying to determine what events are similar enough to be counted as the same. You will never have a bunch of people taking drugs, or a bunch of students being tested, or a bunch of cities implementing policies, that are exactly the same. Much of the meat of your job is in trying to determine what to control for so that, when you're done, they're similar enough to give you back a meaningful answer.
When it comes to predictions, the best you can do is try to train, and then test, on things you think are sufficiently similar. The whole point of cross-validation is to examine how internally consistent your data and model are. If you can train on some, and accurately predict on the rest, a solid interpretation is that the two sets of data are "similar enough." (Assuming away the other enormous part, that your model is correct.) So for observed data, you can assess predictive accuracy with cross-validation.
But for the unseen future, the best answer to your question is just "For the predictions to be correct, you have to assume that tomorrow's weather is drawn from the same distribution as all the weather on which the predictive model was fit." And any question of how close becomes dependent upon a particular model, and preference.