By this token, classic Doom is already non-Euclidean because of teleports.

What people often want when they ask for non-Euclidean is a curved space, like, well, the surface of a globe. Or maybe a torus. Or space being infinitely repeated while looking flat (also much like a torus). Some kind of hyperbolic geometry could be fun but likely too mind-boggling.

(The ultimate in non-Euclidean worlds I've seen so far in games is the paradoxical space of Monument Valley.)

Just like "irrational number", which is not any number that is not rational -- we still assume that it is a real number, so i or Aleph-Null are not considered irrational.

If we include teleport points in our metric, we no longer have a manifold (if teleports are one-way, not even a metric space). Similarly for Mountain Valley, or affine/projective manifolds that can be found in some games, it is wrong to call them Euclidean or non-Euclidean.

Anyway, originally "non-Euclidean" was just the hyperbolic plane (the only geometric structure which satisfies all the Euclid's axioms except the fifth), and then (depending on whom you ask) it was extended to also mean other situations which are different but similar in nature -- so spherical geometry (which is just the opposite), Nil/Solv geometry (which also play with parallel lines), but not cylinders, affine manifolds, teleports, taxicab metric, and so on (they don't really do anything interesting to parallel lines).