For the analogue question of Rogier Brussee we got some nice answer in Euclidean n-space. Which of them can be applied also in hyperbolic $n$-space?
Angles are computed in the tangent space of an intersection point. A tangent space is Euclidean, and the method used for Euclidean space will give the answer there.
Intuitively, a tangent space is a very small neighborhood of a point scaled vastly outward. The scaling makes the neighborhood look very much like Euclidean space.
The tangent space is the limit of such scalings.
It is not clear that the angle measure which I will get with this method is independent from the choice of the intersection point. It is also possible that there is no point of intersection. In this case there is no possibility to give a natural definition for angle?
Use the Poincare Unit Ball model. If two k-planes intersect, then you can translate via isometry any point of intersection to the center of the unit ball. Hyperbolic k-planes through the origin coincide with Euclidean k-planes, and have the same intersection angle at any two points of intersection. This is a conformal model, meaning it gives correct measurement of angles.
I don't know a definition of angle for disjoint points, but can see some possible candidates. For example, extend each to larger planes that contain the shortest geodesic connecting them.
Rational trigonometry is the proper framework for such questions, although they need to be reworded in terms of spreads. In fact for higher dimensional subspaces of a Euclidean or non-Euclidean space, there is not just one spread between them: for example between two 2-planes in 4 space there are two spreads which are invariants. To learn about rational trig applied to hyperbolic geometry, check out my YouTube series on WildTrig and then UnivHypGeom at my channel Insights into Mathematics, or my papers on Universal Hyperbolic Geometry I, II, III and IV (latter with A. Alkhaldi) on the ArXiV or at KoG.
I would suggest considering the hyperboloid model in the Minkowski space. Then the angle between hyperbolic k-subspaces equals the angle between linear (k+1)-subspaces they span, and there you can use your euclidean technique, but replacing the euclidean scalar product with the Minkowski one.
Compare this to the angles between "great subspheres" in an n-sphere, there one proceeds similarly.
As for the angle between disjoint subspaces, I guess it will be a complex number with the imaginary part the distance along shortest geodesic and the real part the angle Joel Hass is mentioning in his answer. At least, in the hyperbolic plane the distance between ultraparallels (times i) is the right analog of the angle between intersecting lines.
Thank you for the nice answers! Now I think that the angle and the distance of subspaces can be defined by a common complex number, which real and inmaginary parts give the respective measures.
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