AFAIK, the Chromecast protocol is a Google “thing”.
You can only use it when using a Google environment. Which can be either a Google/Android phone or a Google web browser.
Plex(amp) running on Windows is neither.
If you had a device which can run Plexamp, you could use Plex’s own protocol, named Plex Companion. Plexamp on the Raspberry Pi
This one has the advantage that you don’t need to keep the controlling app running, once the play queue has been started.
