There is a solution for this when it comes to the Push API spec (not for Safari though). When a notification is received, the browser's Service Worker can send

There is a solution for this when it comes to the Push API spec (not for Safari though). When a notification is received, the browser's Service Worker can send an "I got this" webhook event back to the sender. The sender can then decide if they wish to send a message to other browsers that user may have installed. Note that this does also require a user to be logged in to the site so they can be identified across both browsers.