The site icon is only displayed on the panel for bookmarks that don't have a name. To display the favicon on the bookmarks bar instead of the site name:
Make sure the bookmarks bar is turned on. If the bookmarks bar is turned off, use the keyboard shortcut Ctrl + Shift + B (Windows) or ⌘ + Shift + B (Mac OS) to turn it on.
Right-click on the bookmark.
For more information, see the section Bookmarks.