More information about content in Link field autocomplete dropdown?
The autocompletion matches are provided by EntityAutocompleteMatcher::getMatches. The format is hardcoded as $key = "$label ($entity_id)". Label and entity id come from selection manager, which packages entity id and label pairs in an array keyed by entity bundle, so that could be used as a source for adding content type into the results. You'll need to override system.entity_autocomplete route's controller to get to your own matcher.

The same controller is used for rendering the selection dropdown though it does not display the entity ids, only labels (in web/core/misc/autocomplete.js). Overriding that is another thing you'll need to do.