To be clear, the universal cover property is mostly true; there are a few places where portals are enabled/disabled at runtime (in particular the places where one goes back and something changes in the previous place), and thus the topology is actually changed. In most cases the property is actually true, and things actually do change when you go around the hole (e.g. if the annulus contains a platform, you would be able to hold on to this platform and get a second one when you get to the “same” place again, and the existence of both objects together proves that the old and new place is not fully identical).
But you are right, the proper word, to be exact, is topology, although “non-euclidean topology” isn’t quite the right term either. https://hyperbolica.fandom.com/wiki/Fake_Non-Euclidean_Geometry is a quite good description.
HOWEVER, it isn’t Euclidean geometry either - e.g.
A list of things that have been mistakenly claimed as violating Euclid’s 5th postulate “for any line A and point B, there is exactly one unique line that crosses B but not A” is here:
is actually violated a few times accidentally, although not by every use of portals. Furthermore, from Euclid’s axioms, it follows that a line segment is the shortest path between two points, which is absolutely untrue in all “R^n with portals” worlds due to a trivial counterexample.
Feedback taken though, I will look for a better way to describe this. Thanks!