The StackExchange network depends on browser cookies to remember the login state. Most StackExchange sites use *.stackexchange.com as domain, AskUbuntu is one of many exceptions using its own 2nd level domain (askubuntu.com).
The use of different privacy tools becomes more and more common. Unfortunately these mechanism tend to interfere. If this happens, one can temporarily disable all add-ons/extensions, to find out if one of them is responsible for the problem. If the problem disappears, one can enable the suspicios ones one-by-one to find the culprit.
In my case, I am making use of a browser add-on called "Cookie AutoDelete". This add-on deletes all cookies when completely closing the browser, which of course was the problem.
The remedy is to make a white list entry for askubuntu.com in the preferences of the add-on. In my case, the corresponding entries for *.stackexchange.com were already there. I had created them earlier and forgot about it. Thanks @ByteCommander to hint me in the right direction.