Just to wrap things up.
It wasn't TMS' fault, it wasn't my fault (exactly), but it was the *@%!$ FastReport FMX (Delphi Edition installed via GetIt).
I have the paid version but only for VCL.
And like usual I installed the Delphi Edition which can not be divided in VCL and FMX and installed the paid version over it.
I don't know, what the problem was in detail, but when I uninstalled the GetIt FastReport not only the issues with FastReport itself vanished but I was also able to just click-install the XData.bpl.
Thanks for helping, Wagner!
Don't you TMS guys want to start a FNC report manager?
Yeah, I know FlexCel and no, it is not a real alternative.