A person's identity is only a matter of mental state if it pertains to a particular category* of people, and then only as far as such categorisation is useful as a proxy for said mental state, for example to predict that person's behaviour and preferences. Whether a person's identity is that of a particular individual person is a pure matter of fact, so Craig Wright will never be Satoshi Nakamoto regardless of whether he or anyone else believes it or not.

*IsFoxpupStillAVixen.tk