That'd require a whole other push system operating on top of the normal iOS push system. Web app users would have to go through some Google-hosted push clearinghouse that wrapped some new format within the existing Apple push protocol, and it'd only work for Chrome.
[EDIT] Web app developers, that is—users of the push system. And it means when end users have 20 sites sending them pushes that they don't want, they'd have to figure out how to disable them in Chrome's settings, rather than their normal system push settings. It'd be a total mess.
Isn't this literally how it's done with every browser on earth? I'm using Linux. Websites don't use DBus and notify-send, they use native Web APIs to talk with the browser. The browser then uses the OS notifications API to translate them, and yes, it manages all the subscriptions too.
[EDIT] Web app developers, that is—users of the push system. And it means when end users have 20 sites sending them pushes that they don't want, they'd have to figure out how to disable them in Chrome's settings, rather than their normal system push settings. It'd be a total mess.