Does mobile have a comparable feature? Native Push Notifications are an entirely different beast; afaik, there is no way to have browser type notifications like this.
Chrome on Android does support push notifications from the browser. Facebook makes use of this and you can actually get push notifications from Facebook without using their app. Facebook requests in the Chrome browser to use notifications and then you can receive them without even having the browser app open.
For push notifications, yes, but these are "browser / desktop" notifications, which only show up while the application is running. It's possible this works on mobile with the app running in the background, but I haven't seen it before.