Load either the current public release, 9.26.0.2648, or the current beta release, 9.27.0.2795.
On my Shield, running 9.27.0.2795, “Advertise as player” is in Plex App Settings → Advanced.
When casting/flinging from another Plex client, the Plex app on the FireTV must be running and in the foreground for it to be visible to other devices on the network.
I’m not sure if the same is true for Alexa. I do not use that feature.