From your screenshot, it looks like you're using a Mac.
Chrome routes all notifications on the Mac through the native notification center. This has a few side effects:
1) Notifications are automatically hidden after several seconds
2) Notifications cannot set the icon (all notifications generated by Chrome are automatically given a Chrome icon by the Mac notification center).
It is not possible to hide the text that displays the domain of the site, because otherwise people could use notifications for phishing (for example, I could display a notification that says something like "You have a new friend request" that when you click on it actually takes you to
evil.com with a fake facebook signin page).
Unfortunately, on the Mac, the notification center doesn't have a particularly good way to display this type of domain information so it's just stuck in the middle of the notification. On other platforms, we have more control over how notifications are displayed, so it looks a little less clunky (we display the domain on a thin header bar).
-atw