Building a feature something like SSO

Currently on my app a user can create his account from the web, sign in either from the web or the smartphone. When he sign ins from his mobile app, his user id along with this device details for sending push get uploaded to a different app of mine which handles the sending of the push notifications.
So currently whenever something happens on the website, all the users interested or following a topic then get a push notification about the activity on their phones.

What I was thinking is extracting this logic such a way that any website could use my apps signin and signup feature just like facebook or google SSO such that whenever a user logs in their website and something happens I can then do the work of sending the push notifications.

This would also involve enabling the SSO on their smartphones.

I got this idea because for someone new to mobile and web, it would be very tough to understand how push notifications work and then let xmpp also work with their smartphones.