One approach towards conics classification in hyperbolic geometry is to start with Bertrami-Klein model, place a normal Euclidean conic in there and classify it according to the intersections with the absolute. This leads to a handful of nice curves and for the most part, I can imagine how they look.
For example, an elliptic parabola (one double contact with the absolute and two imaginary intersections) would look roughly like an ellipse near its vertex, and it would more and more approximate an equidistant curve as we go towards the infinity along it -- all its points would stay within a certain distance from its axis. Various hyperbola-like curves would have asymptotes so I guess that far enough from their "interesting regions" their curvature would change very slowly and they would look just like straight lines.
The one curve where I still have a problem is the "semi-circular parabola" -- a curve with triple contact and an intersection with the absolute. This curve has a single branch, but no axis -- since one end of the branch "ends" in an intersection with the absolute and the other one in a triple contact, I am not sure how this curve would actually look to an observer in the hyperbolic plane.
Let's create a mental picture in projective geometry first. Pick $x^2+y^2-z^2=0$ as the fundamental conic of your hyperbolic geometry. Pick $[1:0:1]$ as the single intersection and $[-1:0:1]$ as the triple intersection. One degenerate conic satisfying these would be $y(x+z)=xy+yz=0$. But any linear combination of these two would have the same contact pattern, e.g. $x^2+y^2-z^2+xy+yz=0$.
I did a quick experiment on this curve using Cinderella. Here is what this looks like in Beltrami-Klein and Poincaré disk:
Looks to me like the left side has properties that feel a bit like a horocycle (which would have a quadruple point of contact), while the right end would have asymptotic behavior more similar to that of a line.
Of the four common tangents a pair of conics has in general, three would be the tangent in the left triple point, while one would be a real common tangent somewhere on the right. These two intersect in a point outside the fundamental conic, so using that definition of foci, the sole well-defined focus of this conic would be an ultra-finite point.