The following two answers of a little different question helped me solving this issue plus the following steps, so thanks to these users @leozera and @Tad Lispy :
Two subsequent answers
(Please keep in mind, I cannot tell if these preceding steps are really necessary, because my issue now is solved. And especially the next steps take a significant amount of time, so feel free to start from the bottom before continuing with the steps despcribed in the link respective concluded in the following two steps).
To conclude:
Followed both answers and set environment variables in /etc/environment :
GOOGLE_API_KEY=mygeneratedkeyonGoogleCloud
GOOGLE_DEFAULT_CLIENT_ID=mygeneratedclientidonGoogleCloud
GOOGLE_DEFAULT_CLIENT_SECRET=mygeneratedclientsecretOngoogleCloud
which were automatically embraced by double quotes when exported.
I activated all API's, according to:
Chromium.org API-key Howto
With the build-in chromium even after reboot, nothing changed.
Finishing 2 steps:
But then I signed into askubuntu.com via my google account. After that I was logged in also in to my google account in the chromium and could turn on the sync! Password completion works. More to test when I continue working.
This work took a full morning to complete and test.