Many thanks for the answers and suggestions.
I have now created separate search modules for both languages, this can be made visible in the options only for single languages.
There I also set the additional filters without prefix "filter~" for the respective language (then they work as hoped).
Always happy to read my answers were useful. ?
While it could have been simpler without the issue you described, I found handy having duplicate "specialised" modules especially in multilingual for filtering and positioning. (I often need right to left)