Extending the Audio Shortcode

I would use the filter wp_audio_shortcode:

/**
 * Filter the audio shortcode output.
 *
 * @since 3.6.0
 *
 * @param string $html    Audio shortcode HTML output.
 * @param array  $atts    Array of audio shortcode attributes.
 * @param string $audio   Audio file.
 * @param int    $post_id Post ID.
 * @param string $library Media library used for the audio shortcode.
 */
return apply_filters( 'wp_audio_shortcode', $html, $atts, $audio, $post_id, $library );

It seems that $audio is not a string as stated but an object of type WP_Post, so you could use $audio->ID to get the meta values.

function wpse_163324_audio_filter( $html, $atts, $audio, $post_id, $library ) {
    // Use something like
    // $meta_values = get_post_meta( $audio->ID, 'your_audio_metas', true );
    // to get the meta values and add them to the var $html like
    // $html .= '<div>your_content</div>';
    return $html;
}
add_filter( 'wp_audio_shortcode', 'wpse_163324_audio_filter', 10, 5 );