That file did not exist, but files with the same name but a temp ending did. Renaming one did not help.
But then I deleted all (app, container files) and reinstalled everything from scratch and noticed the file ~/Library/Cookies/Cookies.binarycookies did not exist.
Thought once, then opened a website in Safari - and the file came into being. I immediately had Safari enabled.
The file is indeed the crux and it is created by actually using Safari.