Additional features, settings, or issues not covered by specific Microsoft Teams categories
Well, resetting Teams actually showed some changes. English spell checking is working now, but German isn't anymore. Now my German texts are completely underlined in red. Also, Teams is the only App that displays English date and time format instead of German, although "Country and region" and "Regional format" is set to German in the Windows settings.
My conclusion is, that multiple languages are simply not supported in the "New" Teams. For the old version, everything worked fine.