Link custom post type to users membership

If I can understand what you are looking for, you can create a simple function, that will fire when a new of the CPT is created and update a post_meta eg: "user_controler" for that.

Later, you will need to filter, the posts on the post.php, to show posts only related to the current user.

add_action("save_post", function ($post_ID, $post){
    $post = get_post($post_ID);
    $user = get_curent_user();

    if($post->post_type === "my_cpt" && !get_post_meta($post_ID, "my_meta_key")){
        update_post_meta($post_ID, "my_meta_key", $user->key);
    }
    //$user->key matches what you want to kip as link.
});

To filter the posts on the post.php page you can:

add_filter( 'posts_results', 'my_posts_results_filter' );
    
function my_posts_results_filter( $posts ) {
    global $my_global_condition;
    $user = get_curent_user();
    $screen = get_current_sreen();

    $filtered_posts = array();
        
    foreach ( $posts as $post ) {
        if (
        $post->post_type === "my_cpt" && 
        $screen->ID === $my_cpt && 
        get_post_meta($post_ID, "my_meta_key")=== $user->key
        ) {
                // It's allow :).
                $filtered_posts[] = $post;
        }
    }
    return $filtered_posts;
}

You can also use post_where filter.

You can also filter post simple base on the author property.

With .