Can the device also be set up on a slope / on a mountain?

Yes, that's entirely possible. In this case, the posts pointing to the mountain must be concreted in a little deeper than the lower posts on the “valley side”.
Of course, this can reduce the usable height of the poles somewhat, which should be taken into account in advance when planning.

It is important that the system as a whole remains vertical and the gymnastics bars remain horizontal.

Here is an extreme example where the customer concreted the posts accordingly.