Answer
Currently it seems difficult for an third-party app to access a "foreign" directory in the "On My iPhone" section, which, in the system is referred to as /Documents/.
However, if using react-native-fs and then declaring const path = RNFS.DocumentDirectoryPath; the path returned will be one to the "On My iPhone" section, aka. /Documents/.
If you now create a folder or file under this given path, with, let's assume: RNFS.mkdir('${path}/myfolder'); ( where ' is a backtick-symbol inside the mkdir() ), you will be able to see that the following path was created:
Under "On My iPhone" you will see a folder with your app's name and icon (not always an icon), and inside it your created folder (myfolder).
This, though, still does not open the opportunity of writing/reading files outside of this RNFS.DocumentDirectoryPath. Will keep looking.