Usually comes integrated in all mobile operating systems, alongside the native apps.
https://developer.mozilla.org/en-US/docs/WebAPI/Simple_Push
At the bottom it lists the support and I see all browsers with "Not Supported" and then this line "This API is currently available on Firefox OS only for any installed applications."
Mozilla used to be about open standards.
https://dvcs.w3.org/hg/push/raw-file/default/index.html
Is there something about that document that is against open standards?
If you want cross-device compatibility, use a mature API like Cordova.
Usually comes integrated in all mobile operating systems, alongside the native apps.