It has no memory, for one.
It has very short term memory in the form of it's token context. Especially with something like Meta's Coconut.
What makes you think that it does know its in a conversation?
I don't really. Yet. But I also don't think that it is fundamentally impossible for LLMs to think, like you seem to. I also don't think the definition of the word "think" is so narrow that it requires that level of self-awareness. Do you think a mouse is really aware it is a mouse? What about a spider?