Background: A finite projective plane has point-line duality, and affine planes lack it. You can get a projective plane by adding a "line at infinity," and this is reversible.

Years ago I stumbled across this in constructing a counterexample to a graph conjecture, and I never found out if it was something people had realized or used before:
We can also recover point-line duality from an affine plane by just deleting one of the parallel classes of lines. <cont...>

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<...> Today I checked that this structure has the axiomatization:
Given any point P and line L, P not on L, there are:
A1: a unique line L' containing P parallel to L, and
A1': a unique point P' on L "parallel to" (i.e. not colinear with) P.
A2: something about nontriviality, I think "There is a point P and a line L, P not on L" is enough.

The smallest nontrivial projective plane is Fano, with corresponding affine plane the graph K₄; the corresponding thing here is the graph C₄.

Anyone hear of this before, or can find something about it? It seemed kinda cool at the time, and I like this axiomatization.

Erm, now it seems I wasn't being careful enough when thinking about this. This axiomatization allows more structures than I want.

I'll think about it some more myself, but any suggestions would be welcome. First, A2 isn't enough to stop the analogue of the pencil: one line containing all points and each point forming its own line. But worse, I can't immediately prove that two points determine at most one line (and dually).

Yeah, that last part actually doesn't follow from the axioms. The k-uniform 2k-point tight cycle hypergraph (2k points on a circle, every set of k consecutive points is a line) satisfies the axioms but isn't of the form I wanted. (The parallel classes of points are of size two: antipodal points.)

@bmreiniger
You'll like this puzzle that I made recently: thingiverse.com/thing:3856293
That axiomatization is an encapsulation of why the game Dobble (Spot-It in the USA) works.

@christianp
I did see your post (in fact that's what reminded me of this), but hadn't gotten around to commenting. But Dobble/Spot-It is a projective plane (less two lines, conjecturally because of printing convenience); in my structure, there would be cards that shared no images, and more images that don't appear together on a card.