How to list pages with thumbnails, adding a “current-item” class

By $page->ID, I guess you’re trying to fetch ID of the current page, right? If so, that’s incorrect. Page itself is a post type so you need to get the ID through $post->ID and make sure $post is accessible otherwise use get_the_ID() to return ID of the current post/page.

Final code:

<?php

$our_pages = get_pages( array( 'sort_column' => 'menu_order' ) );
foreach ($our_pages as $key => $page_item) :
    $class="";
    if($post->ID == $page_item->ID)
        $class="current-item";

?>
    <div class="menu-item <?php echo $class; ?>">
        <a href="https://wordpress.stackexchange.com/questions/272111/<?php echo esc_url(get_permalink($page_item->ID)); ?>" class="menu-item-clicker">
            <span><?php echo $page_item->post_title ; ?></span>
        </a>
        <?php echo get_the_post_thumbnail($page_item->ID,'full'); ?>
    </div>
<?php endforeach; ?>