Not really. The input and output is already natural language. That is already "anthropomorphizing".
That is, if this is the bar for anthropomorphization its already happened.
Telling the model to "believe in itself" is just stochastic manipulation that has shown enough reliability to be a recipe to make it keep going.
It's only actually anthropomorphizing if you forget it's a trick and think it's a real person.
There is nothing distasteful about it. If people get confused that's on them. They wouldn't be very useful if you couldn't just talk to them. That's kind of the whole point. Otherwise you can just go back to coding by hand. Telling it to believe itself is just input that happens to work. This probably tells us more about human nature than you realize given the corpus on which it is trained. It obviously doesn't mean anyone actually thinks it's a person.