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Based on @Szabolcs Páll's answer I created this utility class with helper methods to be able to get posts of type by menu order and get the next and previous post by menu order as well. I've additionally added conditions to check if the current post is the first or last post to get the last or first post respectively.

For example:

// $currentPost is first by menu order
getPreviousPostByMenuOrder($postType, $$currentPost->ID)
// returns => last post by menu order

// $currentPost is last by menu order
getPreviousPostByMenuOrder($postType, $$currentPost->ID)
// returns => first post by menu order

The full class:

class PostMenuOrderUtils {
    
    public static function getPostsByMenuOrder($postType){
        $args =[
            'post_type' => $postType,
            'orderby' => 'menu_order',
            'order' => 'ASC',
            'posts_per_page' => -1
        ];

        $posts = get_posts($args);

        return $posts;
    }

    public static function getNextPostByMenuOrder($postType, $postID){
        $posts = self::getPostsByMenuOrder($postType);

        $nextPost = null;

        foreach($posts as $key => $value) {
            if($value->ID == $postID){
                $nextPost = $posts[$key] !== end($posts) ? $posts[$key + 1] : $posts[0];

                break;
            }
        }

        return $nextPost;
    }

    public static function getPreviousPostByMenuOrder($postType, $postID){
        $posts = self::getPostsByMenuOrder($postType);


        $prevPost = null;

        foreach($posts as $key => $value) {
            if($value->ID == $postID){
                $prevPost = $key !== 0 ? $posts[$key - 1] : end($posts);
                break;
            }
        }

        return $prevPost;
    }

}