Some of the changes in this PR are extracted from https://github.com/discourse/discourse/pull/17379.
Similar to the bookmarks tab in the new user menu, the messages tab also displays a mix of notifications and messages. When there are unread message notifications, the tab displays all of these notifications at the top and fills the remaining space in the menu with a list of the user's messages. The bubble/badge count on the messages tab indicates how many unread message notifications there are.