Unfortunately I cannot further edit my original post regarding this topic because it reached its 90-days-limit and has been automatically closed.
But as I today (actually) solved the problem I like to link it hereby so others experiencing similar problems might save some time 
The solution is documented at MacOS: Unable to access microphone when try to call · Issue #1130 · vector-im/element-desktop · GitHub since I experienced the problem finally also with MacOS.
tl;dr Turns out that it wasn't a client side problem at all! The absolutely misleading error message about a non-accessible microphone was due to a misconfiguration of the turnserver settings of my matrix homeserver. Or precisely: Obviously they implemented a stricter interpretation of those values from version 1.11.13 on and the newer clients stopped working. The web clients didn't because I use Firefox and the desktop clients are chromium based and those browsers behave differently regarding turnservers.
