If your geometry has portals, the parallel line postulate will no longer hold. If your portals can have different angles between the entrance and the exit you can straight up cross "parallel" lines, and even if you require them to be oriented at the same angle you'll lose "when given a line and a point exactly one parallel line can be drawn".
Euclidean geometry is an extremely precise geometry, and there is nothing wrong with calling any deviation from it "non-Euclidean". Since Euclidean geometry is a fairly specific point in geometry-space, the only real objection is that calling something "non-Euclidean" doesn't tell you much about what it is, but Euclidean geometry isn't a bad choice for a "default" geometry and at least it tells you it's not that. There is no requirement for it to be continuously curved in some spherical or hyperbolic manner to be "non-Euclidean". Even just permitting a single portal into an otherwise Euclidean geometry would utterly shred Euclid's Elements from top to bottom.
By my examination of the original postulates, not only would portals wreck the parallel line postulate, it in fact ruins 4 out of 5 of them, leaving only that a line can be continuously extended (and that line can still self-intersect, which may in fact violate the definition of a "line" and mean all 5 axioms are busted if you really dug in). It means it is no longer possible to draw a line between any two points (portals can shadow portions of the space such that some points are not connectible), cicles lose a lot of their properties that we rely on (imagine the unit circle, and a portal starting x = .5 and going straight up... they're no longer necessarily continuous), right angles may not be equal to each other any more (portals can change the apparent angles, or give two interpretations rather than one), and of course the parallel line doesn't hold. Putting portals into a geometry fairly comprehensively makes it non-Euclidean.
Euclidean geometry is an extremely precise geometry, and there is nothing wrong with calling any deviation from it "non-Euclidean". Since Euclidean geometry is a fairly specific point in geometry-space, the only real objection is that calling something "non-Euclidean" doesn't tell you much about what it is, but Euclidean geometry isn't a bad choice for a "default" geometry and at least it tells you it's not that. There is no requirement for it to be continuously curved in some spherical or hyperbolic manner to be "non-Euclidean". Even just permitting a single portal into an otherwise Euclidean geometry would utterly shred Euclid's Elements from top to bottom.
By my examination of the original postulates, not only would portals wreck the parallel line postulate, it in fact ruins 4 out of 5 of them, leaving only that a line can be continuously extended (and that line can still self-intersect, which may in fact violate the definition of a "line" and mean all 5 axioms are busted if you really dug in). It means it is no longer possible to draw a line between any two points (portals can shadow portions of the space such that some points are not connectible), cicles lose a lot of their properties that we rely on (imagine the unit circle, and a portal starting x = .5 and going straight up... they're no longer necessarily continuous), right angles may not be equal to each other any more (portals can change the apparent angles, or give two interpretations rather than one), and of course the parallel line doesn't hold. Putting portals into a geometry fairly comprehensively makes it non-Euclidean.