I'm pretty sure Firefox doesn't know how to cast, that's a chrome feature.
Secondly, a chromecast dongle can either be targetted locally by an app (such as chrome) or over the internet via https.
If you are just hosting on your windows laptop, you probably don't have a domain with TLS, yes?
From localhost (the laptop itself), if you run chrome, you can probably cast to your dongle whilst on the same LAN.
If you have one of the newer Chromecasts with the remote, you can simply install the Jellyfin app on it directly, and address your Jellyfin install by IP and port.
Plex uses some fancy redirection work around these limitations, but it relies on an external service that they provide.
Hence why I'm g[l]ad my motorbike doesn't count.
I guess I left my meaning too open to interpretation.