Standard 2D geometries, elliptic, Euclidean and hyperbolic, can be all derived from the same basic idea: start with projective geometry formed by lines and planes through origin in $R^3$ and then put some quadric in its way. $x^2 + y^2 + z^2 = 1$ for elliptic, $z^2 = 1$ for Euclidean, and $x^2 + y^2 - z^2 = 1$ for hyperbolic. This can be rewritten as $1x^2 + 1y^2 + z^2 = 1$, $0x^2 + 0y^2 + z^2 = 1$, and $(-1)x^2 + (-1)y^2 + z^2 = 1$.
I tried to vary the parameters for the projection surface and found some possibilities for hybrid geometries that are no longer isotropic because they contain non-isomorphic lines.
Cylinder $x^2 + z^2 = 1$ leads to a hybrid of elliptic and Euclidean geometry, hyperbolic cylinder $x^2 - z^2 = 1$ to a hybrid of Euclidean and hyperbolic geometry and one-sheet hyperboloid $x^2 - y^2 + z^2 = 1$ to a hybrid of elliptic and hyperbolic geometry.
If we consider elliptic geometry to contain no ideal points, Euclidean geometry to contain an ideal line (line at infinity), and hyperbolic geometry to contain an ideal conic (the absolute), then these three hybrids have a single ideal point, two ideal lines that intersect, and an ideal conic, respectively. The difference between the last one and hyperbolic geometry is that hyperbolic geometry has real points inside the conic and ultra-ideal points outside of it, while the elliptic/hyperbolic hybrid is reversed: its real points are outside the conic.
This is as far as I got -- there should be a way to impose metric on these geometries so that straight lines would be shortest distances between points. I'd be interested in knowing whether someone has studied this further?