The ideas and what a modern reading brings to them can be difficult to disentangle. A more charitable take might be that because emergence is fractal that there are uncountable ways not involving Hegel to arrive at and express a similar underlying idea.
Hegel and Hofstadter seem to be trying to find "spirit". To keep it simple, spirit can be defined by the "I" aka self-consciousness (but it could also be an ant colony, the Church, the State, the biosphere, etc... all the way up to "God").
Hegel tries to notice the becoming of the notion of the Concept (which could be noticed as the basic thought-tool) into (and through) the notion of the "I".
Hofstadter is trying to do something similar but instead of the Concept as the base (or starting point), he uses the Axiom. Hofstadter then tries to discover if "I" (or "spirit") can emerge from a system of axioms.
Please Note: In order to keep it short I gloss over about 1500 pages of Hegel's & Hofstadter's works.