Add new fields to nav_menu_item posts in the WordPress menu editor.

AdminMenusImprover.php

<?php

require_once plugin_dir_path( __FILE__ ) . '/CustomWalkerNavMenuEdit.php';

class AdminMenusImprover {
	/**
	 * Class instance.
	 *
	 * @var AdminTaxonomiesImprover instance
	 */
	protected static $instance = false;

	/**
	 * @var array
	 */
	private static $fields = [];

	/**
	 * Get class instance
	 */
	public static function get_instance() {
		if ( ! self::$instance ) {
			self::$instance = new self();
			self::$instance->do_hooks();
		}

		return self::$instance;
	}

	function do_hooks() {
		add_action( 'init', array( $this, 'setup' ) );
	}

	function setup() {
		if ( ! is_admin() ) {
			return;
		}
		$this->init_fields();

		add_filter( 'wp_edit_nav_menu_walker', function () {
			return 'CustomWalkerNavMenuEdit';
		} );

		add_action( 'save_post', array( $this, 'save_fields_values' ), 10, 2 );
	}

    /*
     * HERE you must add your fields
     */
	static function init_fields() {
		self::$fields = [
			'color' => [
				'label' => __('Color', 'YOUR_LANGUAGE_KEY'),
				'type'  => 'color'
			],
			'badge' => [
				'label' => __('Badge', 'YOUR_LANGUAGE_KEY'),
				//'type'            => 'text',
				//'is_full_width'   => true
			]
		];
	}

	/**
	 * Generator for Id of item meta
	 */
	static function get_meta_key( $name ) {
        $slug = 'FOR_EXAMPLE__YOUR_PLUGIN_SLUG';
		return $slug . '_menu_item_' . $name;
	}

	/**
	 * Get current field value with definite name for definite menu item
	 */
	static function get_field_value( $menu_item_id, $name ) {
		return get_post_meta( $menu_item_id, self::get_meta_key( $name ), true );
	}

	/**
	 * Get result fields HTML for single menu item
	 */
	public static function get_fields_html( $item, $depth, $args ) {
		$new_fields = '';

		foreach ( self::$fields as $name => $field ) {
			$new_fields .= self::generate_field_html(
				$item->ID,
				$name,
				self::get_field_value( $item->ID, $name ),
				$field
			);
		}

		return $new_fields;
	}

	/**
	 * Generate single field html
	 * @return string
	 */
	static function generate_field_html( $id, $name, $value, $field ) {
		$is_full_width = isset( $field['is_full_width'] ) ? $field['is_full_width'] : false;
		$label         = isset( $field['label'] ) ? $field['label'] : '';
		$type          = isset( $field['type'] ) ? $field['type'] : 'text';
		$class         = $is_full_width ? 'wide' : 'thin';

		return "
			<p class='additional-menu-field-{$name} description description-{$class}'>
				<label for='edit-menu-item-{$name}-{$id}'>
					{$label}<br>
					<input
						type='{$type}'
						id='edit-menu-item-{$name}-{$id}'
						class='widefat code edit-menu-item-{$name}'
						name='menu-item-{$name}[{$id}]'
						value='{$value}'>
				</label>
			</p>
			";
	}

	/**
	 * Save the newly submitted fields
	 * @hook {action} save_post
	 */
	public function save_fields_values( $post_id, $post ) {
		if ( $post->post_type !== 'nav_menu_item' ) {
			return $post_id; // prevent weird things from happening
		}

		foreach ( self::$fields as $name => $field ) {
			$form_field_name = 'menu-item-' . $name;

			if ( isset( $_POST[ $form_field_name ][ $post_id ] ) ) {
				$key   = self::get_meta_key( $name );
				$value = stripslashes( $_POST[ $form_field_name ][ $post_id ] );
				update_post_meta( $post_id, $key, $value );
			}
		}
	}
}

CustomWalkerNavMenuEdit.php

<?php

require_once ABSPATH . 'wp-admin/includes/nav-menu.php';
require_once plugin_dir_path( __FILE__ ) . '/AdminMenusImprover.php';

class CustomWalkerNavMenuEdit extends Walker_Nav_Menu_Edit {

	function start_el( &$output, $item, $depth = 0, $args = [], $id = 0 ) {
		$item_output = '';
		parent::start_el( $item_output, $item, $depth, $args );
		// Inject $new_fields before: <div class="menu-item-actions description-wide submitbox">
		if ( $new_fields = AdminMenusImprover::get_fields_html( $item, $depth, $args ) ) {
			$item_output = preg_replace( '/(?=<div[^>]+class="[^"]*submitbox)/', $new_fields, $item_output );
		}
		$output .= $item_output;
	}

}

I have found the right way to make it: we must use this walker to be compatible with other plugins and themes.
If someone needs to rewrite the code using this walker, just leave a comment here, I will provide a new code with the changes.