Echo User Name from User with most comments of today?

You can use this query instead to select comments count grouped by user id.

  SELECT user_id, COUNT( * ) AS total FROM wp_comments WHERE DATE(comment_date) = CURDATE() AND user_id <> 0 GROUP BY user_id

by the way, in your where variable you should use $wpdb->prepare() to assign values with variables ( if you are going to use it in future )

  $where = $wpdb->prepare( 'WHERE comment_date >= %s  AND comment_date < %s AND user_id <> 0', $today, $tomorrow );

This code should work for you:

 global $wpdb;
 $where="WHERE DATE(comment_date) = CURDATE() AND user_id <> 0";
 $sql = "SELECT user_id, COUNT( * ) AS total FROM {$wpdb->comments} {$where} GROUP BY user_id";
 $comment_count = $wpdb->get_results($sql, "ARRAY_A");
 if( !empty( $comment_count ) ) {
   foreach ( $comment_count as $count ) {
     $user = get_userdata($count['user_id']); ?>
      <a href="https://wordpress.stackexchange.com/questions/125950/<?php echo site_url()."/author/' . $user->data->user_login; ?>"><?php echo $user->data->display_name; ?></a><?php
   }
 }