1. They definitely have self-perception to the extent necessary to "know they're adopting a behavior" if there is something there to "do" the "knowing." They ingest their own outputs as a normal matter of operation.
2. We don't know if there's something conscious there to do the knowing
3. Ego doesn't seem necessary either to being conscious or for self-perception
I suspect you're right, but we don't understand nearly enough about consciousness to be confident about any of these questions.
So... Can I worship an LLM and start my own religion? Because we really don't know if consciousness requires X, Y or Z let alone what a a superior all knowing almighty's consciousness consists of.
To me, LLMs look so superior in some ways and maybe they are the gods now that they have manifested themselves as large blobs of floating point numbers when humanity is mature enough to receive them.