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Let's say I have an LLM

prompt: your name is Bob, you're pleasant to talk to.

user: hey Bob.
response: hey user! how are you?

but then later

user: hey Bob!
response: Bob says hello.
user: aren't you Bob?
response: I am an AI that simulates Bob
user: so you're not Bob
response: no. Bob is a person that I act out for you.

so, you see, you now have Bob (who will probably never address you directly again), and you have... the AI that simulates Bob. Which is not the same thing as 'I have Bob'.