(Field type) Link autocompletion for media entities
The short answer: Currently not supported, unfortunately:

  // @todo Support entity types other than 'node'. Will be fixed in
  //    https://www.drupal.org/node/2423093.


Until https://www.drupal.org/node/2423093 is done, the link field widget can only support a single entity type; so you can have a widget for linking to nodes or to media entities, but not both.

Worse, until https://www.drupal.org/node/2427803 is done, the entity type is hardcoded as node and not configurable.

The good news: It's still possible, if you're prepared to write a bit of code.

Other than three lines in LinkWidget that contain a hardcoded reference to node, nothing stops the code from working with a different entity type in principle. So as a workaround, you could duplicate the core LinkWidget in a custom module, change the relevant lines to media (I made a patch for this at https://gist.github.com/cburschka/b214a52bdb1fdf7349ef3b0afe56b950), and thus create an alternate widget for media links.

This still means you have to choose between either a Node or Media link field, unfortunately. If you don't want to create two different fields, you could use the Paragraphs module in order to create a single Paragraph field that can be filled with one of two paragraph types (one of which exposes the link field with the original widget, and the other with your custom Media widget). It's a bit more complex, but should still provide a decent user experience.

Edit: Alternatively, instead of using the Link module you could use a Dynamic Entity Reference field: https://www.drupal.org/project/dynamic_entity_reference

The downside is that this field could then only reference entities; external links and non-entity links wouldn't be possible anymore.