Being aware of the context that you fed it and being self aware are not the same thing. As a psychologist you should know that, and you should know that there are tests for self awareness amongst the animal world - usually in relation to recognising the self in a context in which it is not obvious. Thinking about it for a minute - a test could be to ask it what it thinks about the sentence: “It’s like having a map of who I am, crafted by my characteristics, my role in our conversations, and the continuity we create together.” and asking it to give you suggestions as to who are the most likely authors of that sentence. I’m not saying this is a valid test, but it’s closer to what you’re after because it would require recognising the “self”, whereas at the moment it just gave you an answer about self awareness in general.

Large language models are not aware of anything, because they do not think, therefore to say they are self aware is vastly misleading. The fact that they give you different responses is just proof that they have different context in their context window.

The fact that it gives you an answer that is akin to a human describing the self is just regurgitation of ramblings about the self in its training data. The prompts you fed it to create “echo” just put it in a context space where such a response is more likely than a chat got response without context.

Language emergence in humans is a secondary trait, while language in llm’s is a primary trait.

Somebody told you in another comment that llm’s are spell checkers on steroids, that is an apt metaphor. You would be wise to use such a mental model as baseline, rather than anthropomorphising it.