Is it possible to define which part of the text in which of the indexed text fields matches the query?
No, as far as I know and can tell from the Jira, no such feature exists currently. You can, of course, attempt to highlight the parts of the text yourself, but that requires to implement the highlighting and also implement the stemming according to the rules applied by MongoDB.
The whole feature is somewhat complicated - even consuming it - as can be seen from the respective elasticsearch documentation.