In other words, I would like to use Firebase Cloud Messaging (FCM) from AngularJS to implement real-time chat. If I understand it properly, this is not possible yet -- please correct me if I am wrong.
Currently two-way messaging (receive and send messages) via FCM can be done in Android SDK, iOS SDK, in Chrome Apps and Chrome plugins. However JS frameworks (which fall under "Websites in Chrome") can benefit only from one-way push notifications, as stated here:
Websites in Chrome can implement push messaging through service workers and web app manifests following the Webpush standard.