Microsoft finally has announced support for web push notifications support for Edge. Microsoft Edge Notifications work exactly like Chrome or Firefox push notifications. Starting with Edge 17, developers can the web push API to push notifications. Follow this link for more details on how edge notifications work.
As per the latest update from Microsoft's Developer portal, the support for service worker will continue to grow. Edge already supports 5196 API's out of the 12002 API's defined in the Specifications. Chrome browser, on the other hand, supports as many as 7017 APIs out of the 12,002 API's defined as per the Specifications.
A limited version of this has already been released in a preview mode by Microsoft last month. Here is the tweet that talks about the roll out -
Edge leverages the Windows Notification API platform to pass the notification from the webpage to the Notification platform which will then render and manage notification in a consistent way across all Windows Apps.
Permission and its management function exactly as Chrome, Firefox and Safari. Users visiting the site are prompted for Subscription.
Once the user grants permission, web push notifications can be pushed to the user. These permissions can be managed directly from the parent domain site.
These push notifications can be managed using the Microsoft Action Center. Edge now also supports push notifications on Android.
Here is a snapshot of CanIUse.com that captures the status of support for web push API across browsers and operating systems. [as on 25th Oct 2018]