How to display recent / random posts by its category

OK, so if you want to show 3 random posts from every category that current post is assigned to, then…

First of all you should loop through all terms and get random posts for every term:

$term_list = wp_get_post_terms( $post->ID, 'listing-category', array( 'fields' => 'ids' ) );
if ( $term_list && ! is_wp_error($term_list) ) {
    foreach ( $term_list as $term_id ) {
        // get random posts for given term
    }
}

And currently you get random posts from every category, because there is one minor flaw with your query:

$args = array(
    'post_type' => 'listing',
    'category' => $term_list[0], // <- this works with built-in post categories, not your custom taxonomy called listing-category
    'post_status' => 'publish',
    'orderby'   => 'rand',
    'posts_per_page' => 3,
);

So you should use Tax_Query:

$args = array(
    'post_type' => 'listing',
    'post_status' => 'publish',
    'orderby' => 'rand',
    'posts_per_page' => 3,
    'tax_query' => array(
        array( 'taxonomy' => 'listing-category', 'field' => 'term_id', 'terms' => <TERM_ID> ), // change <TERM_ID> for real term
        // so in your code it's $term_list[0] and in my loop just $term_id
    )
);