Load custom posts with same tag as page

You need to specify a post type in your WP_Query arguments when you need to query any other post type accept the build in post type post.

By default, post_type is set to post, so when no specific post type is set manually by the user, WP_Query will query posts from the post type post

Additionally, caller_get_posts have been deprecated for a very very long time now. If you had debug turned on, you would have recieved a deprecation notice about this. The correct arguments to use is ignore_sticky_posts.

I would also not use the $post global as it is unreliable, for reliability, rather use $GLOBALS['wp_the_query']->get_queried_object(). You can read up on this subject in my answer here

get_the_tags() is also faster than wp_get_post_tags() as the latter requires an extra db call.

One last note, wp_reset_query() is used with query_posts, the correct function to use with WP_Query is wp_reset_postdata()

In essence, you can try the following

$post_id = $GLOBALS['wp_the_query']->get_queried_object_id();
$tags = get_the_tags( $post_id );
if (    $tags
     && !is_wp_error( $tags ) 
) {
    $args = [
        'post__not_in' => [$post_id],
        'post_type'    => 'doing',
        'tag__in'      => [$tags[0]->term_id],
        // Rest of your args
    ];
    $my_query = new WP_Query( $args );

    // Your loop

    wp_reset_postdata();
}