I ve been looking in wikipedia and other sites for "hyperbolic angle", but it is not drawn anywhere. Only an area is shaded everywhere. Is it even possible to draw it?
An angle is formed between two rays that originate at a common point. This is true in both Euclidean and hyperbolic geometries, so the question is more, from a given drawing (or two given rays), how would you measure the angle?
In Euclidean geometry, you can derive an area formula that is $A = r^2 \theta/2$, and so the area is directly proportional to the angle. Simply draw an arbitrary circular arc centered on the starting point of the rays, and there you go.
I must admit not knowing or being able to find the area of a hyperbolic sector as a function of angle, but I suspect (from a similar argument to circular sectors), it would also be $A=r^2 \theta/2$, just that $r$ is constant on a hyperbola.
So the reason areas are shaded is because the areas are directly proportional to the actual angles, and this is the only meaningful way to distinguish between a Euclidean and hyperbolic angle between two drawn rays.
The angle at $(0,0)$ of the shaded area is the hyperbolic angle: