Wordpress - How do I get the name of a menu in WordPress?

On WordPress version 4.9.0 and above you can use

wp_get_nav_menu_name($location)

wp_nav_menu_name for more

You can access the menu metadata using the wp_get_nav_menu_object function

BY NAME:

$menu = wp_get_nav_menu_object("my mainmenu" );

BY SLUG:

$menu = wp_get_nav_menu_object("my-mainmenu" );

The return object as follows:

 Object (
   term_id => 4
   name => My Menu Name
   slug => my-menu-name
   term_group => 0
   term_taxonomy_id => 4
   taxonomy => nav_menu
   description => 
   parent => 0
   count => 6
 )

To display the name:

echo $menu->name;

You can get the name like this, using the menu location, so if the menu is updated or you assign other menu you dont have to update anything here:

$locations = get_nav_menu_locations(); //get all menu locations
$menu = wp_get_nav_menu_object($locations['name_of_the_menu_location']);//get the menu object

echo $menu->name; // name of the menu

the 'name_of_the_menu_location' is the one you use to output a menu using wp_nav_menu

<?php
     wp_nav_menu(array(
          'theme_location' => 'footer'//this value
     ));
?>