"Non-Euclidean" is probably just fine meaning "literally anything other than straight Euclidean". Euclidean geometry doesn't have portals in it. Being truly hyperbolic or spherical is an interesting case but I don't think we have to insist that's the only viable meaning. Adjoining portals to Euclidean geometry by my reckoning breaks 3 of the 5 core axioms of Euclidean geometry... which means by sheer axiom break count, Euclidean geometry plus portals is "more" non-Euclidean than spherical or hyperbolic geometry. (Which may not seem like a terribly meaningful count, but I think would be reflected in what proofs in a portalized space would end up looking like. You'd grow a forest of case analyses and conditions in even the simplest proofs.)

(And it's possibly 4 axioms broken; I'm reckoning "all right angles are equal" as still holding, though I'm not 100% sure a "normal" right angle and a right angle generated where a portal bisects the angle is necessarily "equal" in all relevant ways. It's been a while since I did geometry proofs but the more I ponder it the less sure I am. If not that would leave only "a straight line segment can be prolonged indefinitely" as holding, and unlike in Euclidean geometry a straight line segment may intersect itself arbitrarily often.)

Officially it cannot be called a "geometry" at all if it does not satisfy the other axioms. So it's a "non-Euclidean non-geometry."

What if it satisfies an as-of-yet undiscovered set of axioms?

Axioms are not discovered but decided. A nice set of axioms gives birth to interesting anduseful interactions which you are the one that you discover, although in practice there's is a bit of back-and-forth, as you decides the axiom based on what you are trying to achieve and they get changed or tuned as time pass.

Bonus point if the new axioms play well with existing ones, and are useful to work on our reality.

The math term "geometry" has a certain definition. Things that fit it are geometries, things that don't, aren't.

The gee-whiz "what if" sort of question your asking doesn't really mean anything anymore. In about the mid-20th century math finally came to terms with basically anything you can define being a valid subject of investigation. Arguments about whether things like "imaginary numbers" are "real" and therefore worthy of study are largely gone now. (Whether they are "real" for some definition is now a philosophy question.)

Since axioms define the system you are building, if you want "Euclidean except plus portals" you just need to write them down and start studying them. There are multiple possible sets that describe different systems; for a trivial example, you can assert in an axiom that area is conserved on passing through a portal, or you can write axioms that don't conserve that, and follow the implications from there. Video games have had both kinds of portals. Do portals need to be straight or can they be curved? What do curved portals do to other curves if they pass through them? Does "passing through them" even mean anything (note Euclidean geometry doesn't have "time" in it)? How crazy can I be with the portals? What happens if I define a portal as having a boundary corresponding to a Cantor set [1] and pass a line through it in an axiom system that has "time"? There is no one answer to that question, it depends on the axioms you write down, which may or may not even permit such portals.

Mathematicians will accept any and all of these systems if you write them down. Some may prove to be "uninteresting". Some may prove to have contradictions in them. But it's been a long time since a mathematician would even consider berating you about any of those choices not being "realistic" or something.

And I expect it is very likely some topologist somewhere could hear all these idle musings of mine and say "Oh, yes, you want to go to $TOPOLOGY_SUBDISCIPLINE for that." I don't know what the subdiscipline is or I'd name it, but I'm sure there's something already out there for all this. Since the mathematicians stopped worrying about "realness" they've really spread their wings as as discipline.

[1]: https://en.wikipedia.org/wiki/Cantor_set

Rather than including “portal” as a kind of thing in your space, I would think to just say, “around each point, there is an open ball such that the metric (not the metric tensor, just the d(x,y) thing) is the same as a ball in Euclidean space”.

Oh, and then I guess you could add requirements that for any continuous path along with a, continuous choice of an orthonormal basis of the tangent space, along that path, that uh, transporting small volumes along it, not change the volume?

It can be given an as-of-yet unchosen new name :)

Fair point.

> If not that would leave only "a straight line segment can be prolonged indefinitely" as holding, and unlike in Euclidean geometry a straight line segment may intersect itself arbitrarily often.

This one might not hold either, depending on your point of view. If you take the point of view of someone traveling along a line, that can still go on forever.

But if you're measuring the line, portals can easily prevent it from extending indefinitely, by wrapping it around to an earlier part of itself. It would then fail to be the case that, for any distance, there are two points on the line separated by at least that distance.