I can successfully attach different colors to categories, but the little colored corner on the menu doesn’t appear, why is that?
Those little colored corners are added separately because you may want to add other menu items that are not categories and give them a color anyway.
To make it appear you need to set the desired hex color code in the description field of the menu item as explained in the documentation.
Getting Started > Adding Menus
Note: If you can’t see the description field you can enable it by activating it from the Screen Options at the top right of the “Setting > Menus” admin page.