Except for the string translations I also had translated the main menu. I guess that's not wrong per se. At least it's not explained anywhere I'm not supposed to do that.
What I had forgotten was to associate the Swedish menu with the English. The behavior is still odd considering this, but it works.
Appearance / Menus with Swedish selected:
Menu Name [Menu 1SW] Translations: English Synchronize menus between languages. Language [Swedish] Translation of [Menu 1]
"Translation of" was earlier [--none--]
I won't remove them, but the string translations for the menu are probably redundant now. I noted that the AI didn't even mention menu translation as an option.