How to add item extras to user account menu?

Menu Item Extras extends menu_link_content entity, which is used for MenuLinkContentDeriver provided by the menu_link_content module.
The user module provides its menu link definitions, which are not content entities, so it has strictly defined config fields. As a workaround, you can disable the user module-provided menu items and create a Custom menu item that follows the user account page with the desired fields.