Thanks for your reply!
Both in database and in wp-config.php the locale is set to de_DE.
When I leave my backend admin setting to "Wordpress default", I get a German backend - but I get an English admin bar in frontend. When I opt for "German" in this setting, both the backend and the frontend admin elements are in English (while the option "English" leads to the expected result)!
I tested xili-Language, but installed Polylang before - I had it deactivated when I tested xili. Xili is deleted, also the entry in the options table of my database.
Still I encounter my problem. :-( So what can I do?