Bob’s state might change, but he is still Bob. That would also go for the Ship of Theseus. However you represent it, change is still change, it is still mutability.

We can argue if identity is useful in real life, but most people cling to names and identity, it isn't a controversial subject.

Dynamic inheritance is extremely useful in a programming language, though only a few have it. It is a very OO concept, as are the languages that have explored the concept (e.g. Self, Cecil, among others, even Javascript has this aspect, though not focused enough to use very well).