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;