Where did you get this from? The Wikipedia page you cite clearly disagrees with you:
His influence has led to the current usage of the term "non-Euclidean geometry" to mean either "hyperbolic" or "elliptic" geometry.
It does not say that "replacing the parallel postulate" is an example (it says this is what you do, and there are also kinematic geometries) and also "replacing the parallel postulate" means that you keep all the other postulates. If other postulates are not honoured either, it is in no way closer to hyperbolic geometry.
And I am still using non-Euclidean geometry wider than the Wikipedia line above, to mean "a geometry which is not Euclidean" like you want, i.e., including three-dimensional geometries like Solv and Nil. They are geometries i.e. they stretch the space, while portals do not stretch the space, they change the topology, not the geometry. For example, all triangles will still have angles which sum to 180 degrees, while in hyperbolic geometry, all have less.