I changed quite a few of Siri's settings back and forth, but was unable to fix the problem. Resets and restarts would not help either.
What finally helped was a rather unexpected:
- I switched off the "Dial Assist" function in the "Phone" settings and Siri was able to make calls again.
- I then switched "Dial assist" back on and Siri could still dial without any issues.
I can just assume, that some not otherwise accessible parmeter has been set to a valid value when switching Dial Assist off and on.