How to give each category name its own ACF background color?

The syntax to get the ACF field of a custom taxonomy is:

$var = get_field('name_of_acf_field', 'name_of_taxonomy' . '_' . taxonomy_ID);

so in your case you’ll need to get product_cat ID before and do something like:

add_action( 'woocommerce_shop_loop_item_title', 'VS_woo_loop_product_title', 10 );
function VS_woo_loop_product_title() {
    echo '<h3>' . get_the_title() . '</h3>';
    $terms = get_the_terms( $post->ID, 'product_cat' );
    if ( $terms && ! is_wp_error( $terms ) ) :
    //only displayed if the product has at least one category
        $cat_links = array();
        foreach ( $terms as $term ) {
            $cat_links[] = $term->name;
            $cat_id = $term->term_id;
        }

        $on_cat = join( " ", $cat_links );
        $bgcolor = get_field('kollekcio_szine', 'product_cat' . '_' . $cat_id);

        ?>
        <div style="background-color: <?php echo $bgcolor; ?>">
            <?php echo $on_cat; ?>
        </div>
    <?php endif;
}