As reported in T295621, this notification is one of (if not the only) notifications which directly takes the user away from a MediaWiki project. As such, the default way of marking the notification as read doesn't work.

Users are sent to a URL like https://wikipedialibrary.wmflabs.org/?markasread=4013139&markasreadwiki=metawiki. The query parameters would mark the notification as read if the destination was another Wikimedia project, but don't do anything on the Library Card platform.

We should investigate how to mark the notification as read when clicked.

Per @Legoktm: "Another option could be to add TWL to the Interwiki map and use links like https://meta.wikimedia.org/wiki/Special:GoToInterwiki/google:foo to provide an interstitial and have the markasread query parameters work."