Ok, so the problem is with Minerva Neue. Then I will report it there.
However, I can confirm that language switching is not possible with the skin citizen either. Tested with the current version of the Language Extension Bundle (2023-01-24 MLEB 2023.01).

@Tacsipacsi Thank you for your detailed explanation! That is not an option then.

So it looks like MinervaNeue won't work with the Translate extension unless you use it the way it is used in the Wikimedia wikis.
@Tacsipacsi I'm not really familiar with interwiki unfortunately. Does it make sense to change my wiki to interwiki?

@Jdlrobson maybe T302200 is a similar error but on my wiki the error also occurs on pages that are in site's default content language.
Regarding T64702, I think this has already been tried. It was suggested by @Nikerabbit that I enable $wgPageTranslationLanguageList = 'sidebar-only'. But this does not solve the problem.

@Tacsipacsi that would mean that the problem always occurs with translations in connection with MinervaNeue. But on this page it works with MinervaNeue:
https://www.mediawiki.org/wiki/Help:Navigation

That sounds plausible. I haven't found a way to set that yet - if the question was directed at me.

Thanks for response! By activating the setting $wgPageTranslationLanguageList = 'sidebar-only' , the language selector is now suddenly clickable in MinervaNeue. However, a page appears where you can search for languages, but no languages are listed. Do the supported languages have to be entered somewhere via a configuration setting for MinervaNeue?
And yes, in timeless now a sidebar section with "Deutsch" and "English" is shown, but that's fine.

Yes, I do not use any except the interface language selector. I use the Language Extension Bundle.
With the Translate Extension setting $wgTranslatePageTranslationULS = true;
With this setting when a user selects an interface language via ULS (Universal Language Selector), if the current page is a translatable page or a translation page then the user is also redirected to the corresponding translation page in the selected language.
And this works fine with the timeless skin in the current setup you can find under the provided link.

@Aklapper Ok, what can I do to help solve the problem? Do you need more info? Who is the right person to contact for the problem? Thanks