In Jacobson's Basic Algebra I, in Kaplansky's Linear algebra and geometry and in Artin's Geometric algebra, a hyperbolic plane is defined to be a two-dimensional, nondegenerate inner product space with nonzero null vectors. (I also noticed that Lang went out of his way to define these for alternating forms only, and if I remember right, one of the three other authors did it for symmetric forms only, but I'm not sure at the moment. This may be beside the point.)
When I first heard of these, I thought "oh, there must be a model of synthetic hyperbolic geometry in there somewhere." I thought this was a good guess because, after all, if you fix a point in the synthetic Euclidean plane and look at the transformations that fix it, it looks like the geometry of the regular dot product on $\Bbb R^2$, and the distance even matches the one extracted from the dot product.
But now I'm not so sure the analogous hyperbolic picture is so direct. It's hard to see why the "hyperbolic inner product plane" defined above would be the same as what is going on around a fixed point in hyperbolic geometry. In fact, I have a friend trying to convince me that the two are unrelated. One of his reasons is "in hyperbolic geometry, there aren't two distinguished null lines through points like there are in the inner product space." Finally, it's totally unclear (to me) if there is any connection between the inner product and the distance in hyperbolic geometry.
The fact that both orthogonal and symplectic geometries can have hyperbolic planes adds another layer of complexity that I don't know what to do with.
I'm anticipating one of these types of answers:
Of course there's a useful model of hyperbolic geometry in the hyperbolic plane. You just have to take this subset with these lines and here's how the inner product matches up to the distance like so...
Yeah, they're related, you just need to think of hyperbolic geometry and Minkowskian geometry this way...
No, they're not really directly related. You should just just think of the inner product version separately as Minkowskian geometry...
Of course, I may not be anticipating the best answer possible: I hope to be surprised! :)
Note: I am aware that you can take $\Bbb R^3$ with a metric signature $++-$ and view it projectively to make a model of the synthetic hyperbolic geometry complete with a distance function, but that is not the direction of this question.
I also considered taking the subset of the hyperbolic space consisting of just the points with positive (or just negative) lengths. I wasn't able to convince myself that it was "the right picture," but maybe someone else can convince me.