Adding a login form that concatenates three fields into a username

If you create the user by passing the values entered to a function that combines them, then you can call it from another that calls wp_insert_user. Then you can save the username/pass as needed, and use the same function and form when preparing them for WP to check on log-in.
All of the code below is intended for demonstration only. Can’t stress that enough. Definitely look into nonce and validating registration.

Again, this is ONLY a demo of the idea to point in the general direction.

<form>
    <input type="text" name"value1" />
    <input type="text" name"value2" />
    <input type="text" name"value3" />
</form>

Then we have a function to handle the combining of the entries:

<?php 
function getting_posted_values() {
    if(isset($_POST['value1']) && (isset($_POST['value2']) && (isset($_POST['value3']) {
       $value1 = $_POST['value1'];
       $value2 = $_POST['value2'];
       $value3 = $_POST['value3'];
       $new_value = $value1 . $value2 . $value3;
    }
  return $new_value;
 }

Then a hooked function – (for instance if you used Contact Form 7 to manage the form, you could hook wpcf7_before_send_mail and get access to the post object.) – could call the function below and build the new user:

function build_a_new_user($posted_form) {
    $new_value = getting_posted_values($posted_form);
    $userdata = array(
          'user_login'   => $new_value,
          'user_pass'    => $new_value, //normally: wp_generate_password( 12, false );
          'user_email'   => //note: not sure but may be required?
          'nickname'     => $new_value,
          'display_name' => $new_value,
          'first_name'   => $new_value,
          'last_name'    => $new_value,
          'role'         => 'subscriber'
         );
     wp_insert_user($userdata);

}

wp_insert_user() is documented here.

More on handling custom registration here.

For Logging In

Your form with values 1,2,3 would run getting_post_values again by being called within another function that is hooked to log in somehow. Perhaps the login_form_defaults filter that can edit the args array for the form. The follow is complete speculation as a non-working example.

But the idea is that if you can pass those values to WP’s normal process (perhaps via the authenticate filter mentioned in your linked example?) as the user/pass, you can forgo doing the checking yourself.

function log_in_handler($posted_form) {
    $pass_and_username = getting_posted_values($posted_form);
    $args_array = array(
       'username_id' => $pass_and_username, 
       'password' => $pass_and_username);        
}

It’s definitely an interesting endeavor. Hope that helps.