Get menu object from theme_location

This method looks like what you’re looking for, using get_nav_menu_locations() and get_term():

$theme_locations = get_nav_menu_locations();

$menu_obj = get_term( $theme_locations[$theme_location], 'nav_menu' );

$menu_name = $menu_obj->name;

(See the link for the whole thing wrapped up in a custom function; the above code just highlights the relevant WP functions for getting what you’re after.)