Retrieve Menu name for nav_menu_item

The third argument passed to the filter are the $args used for wp_nav_menu:

function wpse_147499_nav_menu_css_class( $classes, $item, $args ) {
     print_r( $args );
     exit;
}

add_filter( 'nav_menu_css_class', 'wpse_147499_nav_menu_css_class', 10, 3 );

…which gives you:

stdClass Object
(
    [menu] => 
    [container] => 
    [container_class] => 
    [container_id] => 
    [menu_class] =>
    [menu_id] => 
    [echo] => 1
    [fallback_cb] => 
    [before] => 
    [after] => 
    [link_before] => 
    [link_after] => 
    [items_wrap] =>
    [depth] => 0
    [walker] => 
    [theme_location] =>
)

Obviously the values of these vary, and never assume that one will always be non-empty – but hopefully you should be able to differentiate between menus.