quantum mechanics - The Interpretation of $|psi|^2$ and the No-Clone Theorem

The standard interpretation of $|\psi|^2$ is taken as the probability density of the wave-function collapsing in the given infinitesimal small region. The way this probability is interpreted (at least in the text-book by Griffiths) is that if I prepare a large number of identical states and then perform a measurement on each of them then the probability associated with $|\psi|^2$ actually represents the statistical results of the measurements made (individually) on all these wave-functions of the entire ensemble. This seems like a good enough argument to accept $|\psi|^2$ as the probability density of the wave-function collapsing in a given infinitesimal small region even if I have only one wave-function.

But the No-Clone Theorem suggests that it is fundamentally impossible to make two states that are completely identical and thus, to my understanding, it makes absolutely no sense to talk about those identical wave-functions that were used to project $|\psi|^2$ as the probability density. Put in other words, if I can't replicate a wave-function at all then how do I confirm in my laboratory that the probabilities obtained by $|\psi|^2$ actually represents the likely-hood of the collapse happening in a given region?

Edit Is it the catch that the No-Clone Theorem suggests only that a given state can't be evolved to a state identical to another state and it allows the two states being identical if they are so from eternity?

Answer

But the No-Clone Theorem suggests that it is fundamentally impossible to make two states that are completely identical[...]

No, it doesn't say that. It says that it is impossible to build a quantum apparatus (modeled by a unitary operator and a state space for that apparatus) that takes any quantum state as input and then outputs the exact same state, i.e. "clones" it. It does not prohibit the existence of an apparatus that produces a never-ending stream of copies of a fixed state, and it say nothing at all about two states being "accidentally" the same.

If you consider that the crucial feature of bosons is that more than one of them can exist in the same state, the no-cloning theorem in your phrasing would be in outright contradiction to what quantum mechanics actually says.