How does Firefox know I have Facebook messages or notifications if I'm not signed into Facebook?

Did you install any FB apps? Also I think Windows itself has a FB setting possibly in Windows and that might be the one connected that is getting the message. So look at your Windows settings to see if there is such a setting there. I know it has Skype there and I disable that.

This is done via Push notifications that are send via a special Push server and are received in the background, whether you are connected to server or not. These notifications require that you give permission to receive the notification when the website prompt you.

• Options/Preferences -> Privacy & Security -> Permissions -> Notifications

You can see all registered service workers on this page: about:serviceworkers

See also:

• https://support.mozilla.org/en-US/kb/push-notifications-firefox