How to add ‘class’ attribute into shortcode [audio]?

WordPress allows least three options to manipulate the code to your needs.

1. Use a global variable that you increment with @Jevuska ‘s answer.

   global $my_audio_player_count;
   $my_audio_player_count = 0;
   
   add_filter( 'wp_audio_shortcode_class', 'wpse221201_audio_shortcode_class', 1, 1 );
   function wpse221201_audio_shortcode_class( $class ) {
       global $my_audio_player_count;
       $class .= ' my-audio-player-'.$my_audio_player_count++;
       return $class;
   }
   

2. Remove the built-in shortcode and add your own

   remove_shortcode('audio');
   add_shortcode('audio', 'my_audio_shortcode');
   
   function my_audio_shortcode($args) {
       //TODO: Insert your own version of the shortcode
   }
   

3. Use the wp_audio_shortcode_override filter hook to override

   add_filter('wp_audio_shortcode_override', 'my_audio_shortcode');
   
   function my_audio_shortcode($args) {
       //TODO: Insert your own version of the shortcode
   }