I have been thinking, it would be nice if there is a web extension support from Composr CMS. For example, the extension can play a role similar to that of notification feature currently present in the Chrome browser. The social media alert, news feeds and forum notification can be shown to the members who are signed up for a site that uses Composr CMS.

Also, the extension should be supported in all popular browsers like Firefox, Edge, Safari, Opera and more.

Any technical feasibility to achieve so?

Could be done, but someone would need to volunteer/pay for a month or two of work to support all those browsers.
Background notifications are a HTML5 feature available in (I think) all major browsers, probably would make much more sense to implement that first:

Notifications  |  web.dev

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Still a lot of work though.