It seems to me that this has something to do with the inverted (?) color scheme you're using. Generally, the revamped system works like this:
There are two types of messages, inbox and notifications.
We’re fiercely protective of the inbox, making sure only actionable
things directed at you go into it. Comments, answers, a handful of
post notices, ...
Believe me, the Great Big Orange Banner was infinitely more annoying.
 As a mathematician, I'm officially sanctioned one hyperbole per day. This was not it.
 And I'm prohibited Bad Puns. Tough.
This might not be exactly what you are looking but there is a feature for getting emails with inbox notifications from Stack Exchange sites.
In the page for posting questions, https://tex.stackexchange.com/questions/ask, there is a link to the dialogue for inbox notifications. The link is at the mouse pointer in the following screenshot:
You can also reach ...