Everytime a new message is sent in the chat there is an event which is fired to the server so the server can see that event, write that message into the chat (possibly save it to the database at the same time) and tell all clients that this new message appeared in the chat and that they should display it.
Even though you can only see one chat at a time, except party and whisperings, you still recieve messages from all the other chats too, but you can only see the one chat you have open.
All that has to be done is check on the client side if you have the clan chat open and if not make it glow when the server tells the client that a new message appeared.
Apparantly everything that has to be done to implement this notification is already done. The only thing missing is to make the chat do something, which at worst is a UI problem that takes a little bit more effort to fix.
I honestly can't see why a notification for the clan chat isn't there yet.