I disagree with this request. SO is not running a mail service here - the 'inbox' is really a colloquialism for a notification center - something that you read once and move on.
Seeing as there is very little substantial user-made content up there (the excerpts from comments are as close as it gets, and they only go about 75 characters), it doesn't look like there's content worthy to 'read later' there. It may make sense if you didn't have infinite1 time to view them after they've been read, and I particularly don't like a little red box on the top of my screen all the time. Call it whatever you'd like, OCD, Crazy, or insane, but I don't see the usefulness.
I shouldn't have to click on every little notification (especially if it's because of something like a badge that I earned, or a 'thanks' response to an answer) in order to get it out of my inbox, and nor do we need to clutter that interface more with a 'mark all as read' button.
The interface is simple, and it works. Don't mess it up because some people want a mail client in here, it will become clumsy and difficult to manage.