No. Bookmarks live at x.com/i/bookmarks as a list you can scroll. Settings has no “download bookmarks” control. The X bookmarks guide covers what that list is: private, unsearchable as a library, and empty wherever the original post has been removed.
So “export” always means a tool outside that screen. The three that people actually use are the account data download, a browser extension that reads the list, and saving a copy of each post as you go. They do not produce the same file.
Your bookmarks are private to your account, which is also why no outside tool can read them without your sign-in. See are X bookmarks public?