But I still see the release as "Nah und fern" and none of any of four discs has their own title/name.
You will now need to configure the view that you are using to show the 'Set subtitle' tag.
It looks like you are using the Music Explorer view?
I'm not very familiar with that one, so you may want to do some forum searches on that.
But considering the purpose of that view I could imagine Music Explorer perhaps not being able to display custom tags.
Not sure though.
edit:
Not sure how familiar you are with MusicBee already, but if you would use the Album View on your library, the setting for that panel could be accessed here: