How to see that circle?

It is always there when I open the menu to see a notification (using Firefox 38.0 on linux).

I cannot currently replicate this, using Firefox 38.0 on Ubuntu, so more details would be great!
Based on the position of the 'arrow' at the top of the flyout, I think your screenshot is showing the "Messages (0)" tab for Flow notifications (rather than the "Alerts" tab for all other notifications).
This implies that the circle is the "Mark all as read" link (it only shows on the Messages tab), but is displaying strangely (as a circle) for some reason.
It also should not be showing at all, when you have 0 unread Messages.
Here's mine, with 7 unread. If I click it, the link/message vanishes, and is no longer part of the rendered page (it doesn't show up in web-inspector).

See also this (unrelated?) task being worked on this sprint: T72298: C9. "Mark all as read" button can't be found by browser tests

That circle has some effect on hover (see the screencast I added to the description), but I cannot right click on it to inspect the element, so I clicked on the left. I don't see which element is responsible for that circle.

Is this even still relevant with the new ooui version of the popup? Can anyone verify so we can close this or reopen if needed? The entire popup was refactored.