We don't really need such notifications now that we have the updated threads / new posts highlight at the top of the forum. It's not a major issue of course, just a little annoying thing.
Same problem (Firefox and uBlock Origin) and it's been happening for a long time - marking all messages read doesn't seem to do anything, neither does clicking the x boxes either individually or the dismiss all. It occurs infrequently but afaik randomly.
I looked at it a few years ago, and iirc it's an old mod to SMF that does it. Erik can tell us more about it if that's the case, and unless he really loves the add-on I think we could probably do without it, but that's up to him.
edit: my guess is it's Enotify - it's installed (see the bottom of any forum page) and the SMF community forum thread for it has some bug complaints about old post popups. That bug may have not been fixed.
My hunch was it only happened to really old accounts that were/had messages before this was implemented. However King-Salomon has an older account than some who confirmed the issue, so that can't be it.
Maybe the browser? Firefox here, using uBlock and Ghostery too.