It is possible to obtain two PhD degrees (various fields) on the basis of one interdisciplinary PhD dissertation? It is possible to obtain two PhD degrees (various fields) on the basis of one interdisciplinary PhD dissertation? Is this something feasible (in Europe)?

Or maybe there must be two different 'physical' doctoral dissertations?
No, at least the regulations that I am aware of require that the dissertation be a novel contribution to the respective field and that it has not been submitted before.

If you write a dissertation at the crossroads of different fields, you have to decide for a field in which you want to formally graduate and obtain a degree. It may be advisable to look for supervisors from different fields or with an interdisciplinary background.