Not sure self-reference solves the metacognition thing; an ant can pass the mirror test but probably lacks metacognition.
Though I haven't read GEB so I'm not sure how the strange loop thing ties in with either of those.
You should read his later work called I Am a Strange Loop instead of GEB, as this is the author’s preference.
You should read his later work called I Am a Strange Loop instead of GEB, as this is the author’s preference.