Posts alphabetically in a list with clickable links

Use the get_posts() function instead of get_terms(). You will probably want to use it something like:
$cats = get_posts( array( 'post_type' => 'author', 'orderby' => 'title', 'order' => ASC ) );

Displays:
Catchable fatal error: Object of class WP_Error could not be converted to string in /beta/wp-includes/formatting.php on line 433

Maybe must you change this line?:
$url = attribute_escape( get_category_link( $cat->term_id ) );

and:

$name = apply_filters( 'the_title', $cat->name );

Well, the loop will need to be changed to work with the properties that should be returned by the get_posts() function instead of the properties that were returned by the get_terms() function.

Therefore, things like $cat->name[0] or $cat->name would become $cat->title[0] or $cat->title, etc. In addition, rather than trying to use get_category_link( $cat->term_id ), you should use get_permalink( $cat->ID ).

Nothing is displayed…

Code:

<?php
		$list = '';
		$cats = get_posts( array( 'post_type' => 'autorzy', 'orderby' => 'title', 'order' => 'ASC' ) );
		$groups = array();
		if( $cats && is_array( $cats ) ) {
			foreach( $cats as $cat ) {
				$first_letter = strtoupper( $cat->title[0] );
				$groups[ $first_letter ][] = $cat;
			}
			if( !empty( $groups ) ) {
				{
					$index_row .='<ul class="topindex">';
					foreach ($groups as $letter => $cats) {
						$index_row .= '<li><h4><a href="#' . $letter . '" title="' . $letter . '">' . apply_filters( 'the_title', $letter ) . '</a></h4></li>';
					}
					$index_row .='</ul><br class="clear" />';
				} 

				$list .= '<ul class="index">';
				foreach( $groups as $letter => $cats ) {
					$list .= '<li><a name="' . $letter . '"></a><h5>' . apply_filters( 'the_title', $letter ) . '</h5>';
					$list .= '<ul class="links">';
					foreach( $cats as $cat ) {
						$url = attribute_escape( get_permalink( $cat->ID ) );
						$name = apply_filters( 'the_title', $cat->title );
						$list .= '<li><a title="' . $name . '" href="' . $url . '">' . $name . '</a></li>';
					} $list .= '</ul></li>';
				} $list .= '</ul>';
			}
		} else $list .= '<p>Brak autorów.</p>'; ?>

		<a name="cats_top"></a>
		<?php print $index_row; ?>
		<?php print $list; ?>

Post type:

add_action('init', 'my_custom_init');

function my_custom_init() {

	$labels = array(

		'name' => 'Autorzy',

		'singular_name' => 'Autorzy',

		'add_new' => 'Dodaj autora',

		'add_new_item' => 'Dodaj nowego autora',

		'edit_item' => 'Edytuj autora',

		'new_item' => 'Nowy autor',

		'view_item' => 'Zobacz autora',

		'search_items' => 'Szukaj autora',

		'not_found' => 'Brak autora',

		'not_found_in_trash' => 'Brak autora w Koszu', 

		'parent_item_colon' => ''

	);

	$args = array(

		'labels' => $labels,

		'public' => true,

		'publicly_queryable' => true,

		'show_ui' => true,

		'query_var' => true,

		'rewrite' => array('slug' => 'autor'),

		'capability_type' => 'post',

		'hierarchical' => false,

		'menu_position' => 5,

		'menu_icon' => 'https://wp.pabloart.pl/bwa/wp-content/themes/bwa/img/autorzy.png',

		'taxonomies' => array('category'),

		'supports' => array('title', 'editor')

	);

	register_post_type('autorzy', $args);

}