Add classes + taxonomy terms to wp_list_pages() output

Doesn’t WordPress already at the class current_page_parent, current_page_ancestor and current_page_item (see source). The filter you are using is called on 1046.

As for your question, you create the array $content_type and then attempt to make a string out of $the_content_type.

Try:

 function wpse56088_page_css_class( $css_class, $page, $depth, $args, $current_page ){
     $terms = get_the_terms( $page->ID, 'content-type' );
     if ( $terms && ! is_wp_error( $terms ) ){
        $term_names = wp_list_pluck($terms,'name');

        //Add term names to array of classes
        $css_class = array_merge($css_class, $term_names);
     }
     return $css_class;
}
add_filter( 'page_css_class', 'wpse56088_page_css_class', 10, 5 );

Note $css_class is suppose to be an array of class names.