This is essentially a philosophical rather than a physical question. (I don't mean that as a pejorative statement, by the way, just a descriptive one.) There are different philosophical approaches regarding the meaning of probability. Broadly speaking, some people think of probability in a frequentist sense, meaning that probabilities refer to the frequency of outcomes in an ensemble. Other people think of probability as a description of our knowledge, rather than a description of the system itself. In this way of thinking about things, often called "Bayesian," probabilities don't require one to postulate or even to imagine ensembles.
Personally, I think that the frequentist attitude is self-evidently absurd. When I look at the weather report and see that there's a 70% chance of rain tomorrow, I am not required to, and certainly do not, imagine an ensemble of infinitely many Earths, 70% of which will contain rain tomorrow. Rather, this statement simply means that I don't know whether it'll rain or not, but I'll be somewhat more surprised if it doesn't than if it does. E.T. Jaynes's book is a standard (and in my opinion excellent) manifesto in favor of this attitude towards probability.
Weather forecasts are not quantum probabilities (at least not entirely, or in any obvious way), but I don't think that matters. I see no reason I can't regard all probabilities in exactly the same way.
(By the way, the terms frequentist and Bayesian also refer to different sets of statistical techniques. Although that usage is related to the philosophy-of-probability usage, they're not the same. In particular, I am not claiming (nor do I believe) that frequentist statistical techniques are absurd or invalid. I actually use both sets of techniques in my own work, depending on which is more convenient for any given problem.)