In hyperbolic geometry, ideal triangles are all congruent to each other. Convex Ideal quadrilaterals ( a quadrilateral where all 4 points are ideal points) all have the same area (2pi)

In the context of the Poincaré disk model. An isometry of the hyperbolic plane is uniquely determined by mapping three ideal points to their images. That's because that isometry is a Möbius transformation, which is uniquely determined by three points and their images. (Actually if the order of ideal points changes, then you's have to compose this with an inversion in the unit circle, because the Möbius transformation would exchange inside and outside.)
So you can map any three points to any three points in an isometric way. Conversely, if you fix three corners, the position of the fourth uniquely determines the shape of the quadrilateral.
So what parameters can you use to describe that shape? The angle between the diagonals does seem like an intuitive choice. Coming from a background of projective geometry, I'd probably pick the cross ratio of the four ideal vertices. You can compute this e.g. as $\mathrm{\lambda <!--\; \lambda \; -->}=\frac{(a-<!--\; -\; -->c)(b-<!--\; -\; -->d)}{(a-<!--\; -\; -->d)(b-<!--\; -\; -->c)}$ with $a,b,c,d\in <!--\; \in \; -->\mathbb{C}$ having absolute value $1$. Both approaches have benefits and drawbacks.
One thing worth considering is whether you allow for self-intersecting quadrilaterals. If two edges may intersect, then the diagonals would not intersect. Extending them to infinite lines, one could see them intersect in the Beltrami-Klein model, forming an imaginary angle of intersection. So you may want to forbid self intersection, or allow for complex angles. Cross ratios cover the self-intersecting case using real numbers (and $\mathrm{\infty <!--\; \infty \; -->}$ for the special case of two certain points coinciding). To forbid them you could restrict the value of the cross ratio to $1<\mathrm{\lambda <!--\; \lambda \; -->}<\mathrm{\infty <!--\; \infty \; -->}$ for $(a,b,c,d)$ cyclic in that order, or to $0<\mathrm{\lambda <!--\; \lambda \; -->}<1$ for $(a,b,d,c)$ cyclic in that order.