I learned from Emil Artin's book Geometric Algebra that the standard incidence axioms of affine geometry (two points determine a unique line, parallel postulate, no three collinear points exist) together with (specializations of) Desargues' theorem are sufficient to translate theories of affine geometries into theories of modules over division rings.
Artin carries out explicitly the attractive construction of defining an appropriate notion of translation (a map $\tau$ with no fixed points such that the line $PQ$ determined b $P$ and $Q$ is parallel to the line$\tau(P)\tau(Q)$ determined by $\tau(P)$ and $\tau(Q)$), showing that the translations form an abelian group, and then showing that the endomorphisms of this group that fix the directions of the translations (i.e. that fix the pencil of parallel lines determined by any/each of the lines $P\tau(P)$) actually form a division ring. Intuitively, the elements of this division ring are the scalars by which translations are scaled. Then the abelian group of translations turns out to be a 2-dimensional module, and since part of Desargues' theorem states that there exist translations between any two points, this successfully coordinatizes the affine geometry.
If I have understood correctly, adding an ordering to the geometry translates into an ordering on the underlying division ring (hence making it a subfield of $\mathbb R$); distance translates into a norm on the module; angles translate into an inner product, so the above successfully models Euclidean geometry (keeping in mind the choice of the three non-collinear points: $(0,0)$, $(1,0)$, $(0,1)$) as $\mathbb R^2$.
My understanding of why what Artin does actually works is that the properties of translations crucially depend on the "flatness" of affine space, which is encoded by the parallel postulate (the parallel postulate seems to allow a kind of transport of incidence data around one point to another, it seems, and Desargues's theorem as used by Artin guarantees that all such transports are coherent with one another).
Clearly, this is not true with hyperbolic geometry as it negates the parallel postulate. Nevertheless, I've read on numerous wikipedia pages that this negation is the only difference between the standard axioms of (plane) Euclidean geometry and (plane) Hyperbolic geometry. What I am curious about is whether it is possible to determine from primitive synthetic notions (points, lines, betweenness, congruence) the fact that hyperbolic geometry is a 2-dimensional Riemann
surface manifold with constant negative curvature using a construction similar to the one I've read in Artin's book. Obviously, such a construction will be more sophisticated, as it would have to essentially select a coordinate chart around every point, together with transition maps (Artin's construction seems to construct one global chart for the whole of affine space).
Any references, explanations, or corrections would be much appreciated. I should perhaps mention that my nebulous endgoal is to understand hyperbolic distance, but I believe this approach, if feasible, ought to shed much intuition about (plane) hyperbolic geometry.