This sounds a little snarky, but I mean it sincerely. Unfortunately, I'm not sure how to define the notions of "approximate topological-ness" or "approximate g

This sounds a little snarky, but I mean it sincerely. Unfortunately, I'm not sure how to define the notions of "approximate topological-ness" or "approximate geometric-ness" of an order in any rigorous way, so I am reluctant to answer the question. I would agree that trying to fit the order on the surreal numbers into an existing geometric, or even probably topological, intuition is likely to fail. For me, the order is in some sense a combinatorial construction; but I don't know any way to make that rigorous other than that the definition is game-theoretic, and I think of game theory as a particular kind of combinatorics (as probably no specialist in either field ever would).