When adding media to content, how can I search for the media by tag or term?
As Kevin said, you need to install the Entity browser module

Optional module: I personally also use Entity Browser Enhance(d|r) for a much better usability when selecting media.

Go to /admin/structure/views/add and create a view showing Media items. On the next screen add a display type Entity browser to your view. You must add the special Entity browser bulk select form field to this view. Configure the view with additional fields and filters of your own choice. E.g. if you need taxonomy tags, add an exposed filter for your tags field.  

Under Configurarion &gt; Content Authoring &gt; Entity browsers, create a new entity browser for your media. Configure as you need, and in the last step add the previously created view as widget plugin.
(Beyond the selection view you could add multiple widgets here like DropZoneJs for in-place uploading or Inline Entity form for in-place creating of new nodes).

Go to Structure &gt; Content Types, and edit the form display of your node. Change the widget of the media field to Entity browser and select your previously created browser.

There are quite a lot of configuration options on the way, but this is a basic outline to get a media browser like this:
(The search part can contain any/multiple exposed filter like a "normal" view)


I recommend this tutorial for additional information
https://www.electriccitizen.com/blog/media-libraries-drupal-8
(but I personally would not use Inline entity forms for media, better stick with Entity browser for selection and DropZoneJS for in-place uploading)