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Kind of? Keep aside quantum mechanics for a second. In any classical experiment that has random outcomes, would you say that the probability distribution is a property of a single system or a bunch?

You can only deduce a distribution from repeated measurements. But most physicists would have no problem talking about a single experiment having many possible outcomes, governed by a probability distribution. It's almost a philosophical question about whether probability means anything in single systems.

It's the same way in quantum mechanics. The effects of entanglement can only be discerned if you take repeated samples. But we still feel okay talking about single systems governed by such entanglement.