Do conic sections have an interesting generalization to surfaces of constant curvature? Consider a sphere (constant positive curvature) $\mathcal{S}$ centered at $O$, as well as points $A, B \in \mathcal{S}$. For any real constant $0 \leq \theta \leq \pi$, define $$ \mathcal{E}_\theta = \{P \in S\ |\ \angle AOP + \angle BOP = \theta\}. $$
$\mathcal{E}_\theta$ is a somewhat oval-shaped closed curve in $\mathcal{S}$, that is an analogue of an ellipse. Indeed, as the radius of $\mathcal{S}$ grows relative to $\theta$, $\mathcal{E}_\theta$ approaches the limiting case of an ellipse in a flat plane.
Similarly, by considering the difference in angle instead of the sum, we could define an analogue of hyperbolas.
Do these generalizations have interesting properties? Such analogues can be constructed in arbitrary metric spaces.
Most of the properties of ordinary conics come not from a metric, but from the defining equations being of degree 2, in a linear coordinate space. So there is some linear structure beneath and a big symmetry group (like rotations of the circle). Conics belong to projective geometry, which does not have a metric, and algebraic geometry, which tries to not use the metric.
With that warning, the first two things I would look for in metric generalization of conics are
Does the light reflection property of the ellipse generalize to Riemannian manifolds, at points with a unique geodesic to each focus? This is potentially a local differential calculation.
Is there an area-preserving transformation of the hyperbolic plane or the sphere that fixes a metric ellipse and moves lines (geodesics) through the center of the ellipse to other lines? Maybe not, but it should be relatively easy to find out.