I like your example that they become an instance. For the sake of argument say you have Holy Grail AGI™ and you copy it and starting from State t where T is the exact moment you copy it, two instances now exist.
I think they are both "them" but the state change through time as a chaotic process and so since they are now separated they will eventually begin to drift?
This is based totally on nothing but "dude trust me I made it up"
I figure what we're doing in these threads is kind of what sci-fi authors who were way smarter than myself have done for decades: Muse about a topic, put down some ideas maybe some new vocabulary Etc. Then when new technology comes out those of us who are well read can look back and say "A ha! That clever bastard knew it all along!"