Whenever you receive a (noteworthy) badge, answer notification, bounty reminder, etc. you get either a new message in your top-bar global inbox or an achievement notification. These events all used to be sent to the old 'notifications' message box in the previous top-bar design.
Looking at the inbox or achievements drop-down marks those events as read. However, when you visit a Stack Exchange site on mobile, the mobile notifications box still shows all of these events as unread. The same is also true in the other direction, looking at the notifications on mobile first leaves the same messages unread on the main site.
Clearly the mobile UI still uses the old notifications architecture, and this doesn't talk to the new achievements / inbox channels.
Can the mobile notifications be marked as read when I look at them on the main site, and vice versa?