How to customize the built in menu in wordpress?

Try this plugin and this plugin will allow you to add a menu icon for menus.
https://wordpress.org/plugins/menu-icons

You can also use https://codex.wordpress.org/Function_Reference/register_nav_menu to create a custom menu and for displaying menu in your theme you can use https://developer.wordpress.org/reference/functions/wp_nav_menu/

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