Wordpress - How to display categories of my Custom Post Type?

Remove your code from portfolio-type.php that registers post type and taxonomy (line 9 onwards).

Use the following code (in portfolio-type.php) to register the post type "portfolio"

function portfolio_register() {
    $labels = array(
        'name' => _x('Portfolio', 'post type general name'),
        'singular_name' => _x('Portfolio Item', 'post type singular name'),
        'add_new' => _x('Add New', 'portfolio item'),
        'add_new_item' => __('Add New Portfolio Item'),
        'edit_item' => __('Edit Portfolio Item'),
        'new_item' => __('New Portfolio Item'),
        'view_item' => __('View Portfolio Item'),
        'search_items' => __('Search Portfolio Items'),
        'not_found' =>  __('Nothing found'),
        'not_found_in_trash' => __('Nothing found in Trash'),
        'parent_item_colon' => ''
    );
    $args = array(
        'labels' => $labels,
        'public' => true,
        'publicly_queryable' => true,
        'show_ui' => true,
        'query_var' => true,
        'rewrite' => true,
        'capability_type' => 'post',
        'hierarchical' => false,
        'menu_position' => 8,
        'supports' => array('title','editor','thumbnail')
    ); 
    register_post_type( 'portfolio' , $args );
}
add_action('init', 'portfolio_register');

Use the following code (in portfolio-type.php) to register a taxonomy "portfolio_categories" for the post type "portfolio", make it hierarchical (like categories)

function create_portfolio_taxonomies() {
    $labels = array(
        'name'              => _x( 'Categories', 'taxonomy general name' ),
        'singular_name'     => _x( 'Category', 'taxonomy singular name' ),
        'search_items'      => __( 'Search Categories' ),
        'all_items'         => __( 'All Categories' ),
        'parent_item'       => __( 'Parent Category' ),
        'parent_item_colon' => __( 'Parent Category:' ),
        'edit_item'         => __( 'Edit Category' ),
        'update_item'       => __( 'Update Category' ),
        'add_new_item'      => __( 'Add New Category' ),
        'new_item_name'     => __( 'New Category Name' ),
        'menu_name'         => __( 'Categories' ),
    );

    $args = array(
        'hierarchical'      => true, // Set this to 'false' for non-hierarchical taxonomy (like tags)
        'labels'            => $labels,
        'show_ui'           => true,
        'show_admin_column' => true,
        'query_var'         => true,
        'rewrite'           => array( 'slug' => 'categories' ),
    );

    register_taxonomy( 'portfolio_categories', array( 'portfolio' ), $args );
}
add_action( 'init', 'create_portfolio_taxonomies', 0 );

Then use the following code to retrieve taxonomy terms in template files (like index.php)

<?php

$taxonomy = 'portfolio_categories';
$terms = get_terms($taxonomy); // Get all terms of a taxonomy

if ( $terms && !is_wp_error( $terms ) ) :
?>
    <ul>
        <?php foreach ( $terms as $term ) { ?>
            <li><a href="<?php echo get_term_link($term->slug, $taxonomy); ?>"><?php echo $term->name; ?></a></li>
        <?php } ?>
    </ul>
<?php endif;?>

?>

Let me know if you need any clarification.

For this portfolio post categories list use:

<?php echo get_the_term_list(get_the_ID(), 'portfolio_category', '', ', ', ''); ?>