From the user admin panel the red update notification dot within the Site Kit tab appears. When checking for updates there is no Site Kit update available, with the latest version already installed
Original Support topic:

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Users noted in this topic that in an incognito window the notification does not appear.
Closing as this issue no longer persists. Most likely caching issue as per this fixing WordPress article.
Checking from incognito window, clearing cache or temporarily deactivating caching plugins provided resolution for all impacted users.
A user followed up that the issue still persists for them, so reopening the issue for further investigation.
Also have this issue. It's been present since I started using Site Kit a couple of weeks ago. Bugs me because it's distracting.
The notification disappears when there actually is an update for Site Kit and you execute it, only to return shortly thereafter. Minutes, IIRC.
Please note that in the original screenshot the "updates" icon is not present in the top toolbar, but the notification badge on Site Kit is. So, it doesn't seem to be a WP issue and the WordPress.org article linked by jamesozzie doesn't seem applicable.
Don't have caching plugins, not active, and not installed.
Willing to help by providing more information if you need it, and by trying stuff to help diagnose. (I'm now a writer/marketer, but in my previous career I was a developer for 30+ years.)
Update:
Badge disappeared just now when I navigated to "Appearance | Themes". Returned when I went back to the Dashboard. Remains / returns when navigating to Posts. But disappears when opening Pages. AFAICT the badge only shows on Dashboard and Posts, and Site Kit itself of course.
I have the same problem. This is very annoying.
I've also had this same problem since I installed the module a month or so back. Subscribing for updates.
I think this issue created after the latest google site kit update.
please check previous version files and try to update them
I might be missing something, but my understanding is that this notification bubble is not for updates, but for wins and alerts from individual modules.
Curious to hear whether clearing local storage fixes this.
To clear storage in Chrome, go to your WordPress admin, and open the browser's developer tools (press F12). Then head over to the "Application" tab and click on "Clear site data". The screen looks like this:

@swissspidy no that is not the case !
I tried it already but did not help
I've tested this after the fixes from #1781, and they indeed resulted in this working correctly again. The notification bubble wasn't actually incorrectly displaying - the problem was that the actual notification that the counter indicated had _not_ been displaying.
This should be fixed in the upcoming release alongside the notification fixes.
cc @jamesozzie @ernee @LuckynaSan @marrrmarrr