The continuing identity of a person seems to me the same sort of thing as the continuing identity of a physical object, and not particularly problematic. The paradox of Theseus' ship is as easily dissolved as that of unheard falling trees.

I have a day-to-day continuity that I do not have with any part of anyone else since I was born, in just the same way that the chair I am sitting on does not have any continuity with any other chair more recently than when it was assembled on a production line.

To ask whether a person who has undergone a massive change of personality or mental capacity is the "same" person is like asking whether a wrecked car is the "same" car as it was before it was wrecked. To ask whether a person still exists when they are asleep is like asking whether a performance car is still a performance car while it is sitting in the garage.