How to get the number of times an audio file has been played

Yes, the play event fires on the <audio> element, so you could target it eg $('audio.wp-audio-shortcode').on('play', function (event) { //etc });. A number of other events are also available, eg ended, playing, pause.

Here’s some code I’m trialing at the moment that may help you (in “functions.php”):

add_action( 'wp_footer', function () {
    ?>
    <script>
    jQuery(document).ready(function() {
        (function ($) {
            var srcs = []; // Array of sources already sent to cut down on posts to server.
            $('audio.wp-audio-shortcode, .wp-audio-playlist audio').on('play', function (event) {
                // this.src should be the url (guid) of the audio file
                if (this.src && $.inArray(this.src, srcs) === -1) {
                    srcs.push(this.src);
                    $.post( '<?php echo admin_url( 'admin-ajax.php' ); ?>', {
                            action: 'wpse168584_audio_stats',
                            guid: this.src.replace(/\?.+$/, ''), // Remove any query vars.
                            nonce: '<?php echo wp_create_nonce( 'wpse168584_audio_stats_' ); ?>'
                        }, null, 'json'
                    );
                }
            });
        })(jQuery);
    });
    </script>
    <?php
} );
function wpse168584_audio_stats() {
    $ret = array( 'error' => false );

    if ( ! check_ajax_referer( 'wpse168584_audio_stats_', 'nonce', false /*die*/ ) ) {
        $ret['error'] = __( 'Permission error', 'wpfm' );
    } else {
        if ( ! isset( $_REQUEST['guid'] ) || ! ( $guid = $_REQUEST['guid'] ) ) {
            $ret['error'] = __( 'Params error', 'wpfm' );
        } else {
            global $wpdb;
            $sql = $wpdb->prepare( 'SELECT ID FROM ' . $wpdb->posts . ' WHERE guid = %s LIMIT 1', $guid );
            if ( $post_id = $wpdb->get_var( $sql ) ) {
                // Use hex format to save space, 8 bytes for IPv4, 32 for IPv6.
                $ip = bin2hex( inet_pton( preg_replace( '/[^0-9a-fA-F:., ]/', '', $_SERVER['REMOTE_ADDR'] ) ) );
                if ( ! ( $meta = get_post_meta( $post_id, '_wp_attachment_metadata', true ) )
                || ! isset( $meta['plays_ips'] )
                || ! in_array( $ip, $plays_ips = explode( ';', $meta['plays_ips'] ) ) ) {
                    $plays_ips[] = $ip;
                    // If data getting too big, drop off oldest ip (FIFO).
                    if ( strlen( $meta['play_ids'] ) > 1000 ) array_shift( $plays_ips );
                    // Save as string to save space.
                    $meta['plays_ips'] = implode( ';', $plays_ips );
                    $meta['plays'] = isset( $meta['plays'] ) ? $meta['plays'] + 1 : 1;
                    update_post_meta( $post_id, '_wp_attachment_metadata', $meta );
                }
            }
        }
    }

    wp_send_json( $ret );
}
add_action( 'wp_ajax_nopriv_wpse168584_audio_stats', 'wpse168584_audio_stats' );
add_action( 'wp_ajax_wpse168584_audio_stats', 'wpse168584_audio_stats' );

Then to see the “plays” meta data when you go into admin and edit an audio item (it appears in the read-only meta data box on the top right under bitrate):

add_filter( 'media_submitbox_misc_sections', function ( $arr ) {
    $arr['plays'] = __( 'Play Count:' );
    return $arr;
} );