Solution identified!
APN settings. Only restarting them and the phone doesn´t do. They needed to be modified. Weeks of recurring exchange with the network carrier, nothing better.
It seems particularily problematic when the carrier asks for the phone model and hears “Fairphone” and responds “it´s not on my list” + the carrier doesn´t let you access the APN (masked).
It´s total guess work.
Here the links that helped me (Salt/Switzerland) - and thanks @Meaghan - Community angel, for the hint !