I have sucessfully integrated this code in functions.php

//1. Add a new form element...
add_action( 'register_form', 'myplugin_register_form' );
function myplugin_register_form() {

global $wp_roles;

echo '<label for="role">Type de client</label>';
echo '<select name="role" class="input">';
foreach ( $wp_roles->roles as $key=>$value ) {
   // Exclude default roles such as administrator etc. Add your own
   if ( in_array( $value['name'], [ 'Client', 'Entreprise'] )) {
      echo '<option value="'.$key.'">'.$value['name'].'</option>';
   }
}
echo '</select>';
}

//2. Add validation.
add_filter( 'registration_errors', 'myplugin_registration_errors', 10, 3 );
function myplugin_registration_errors( $errors, $sanitized_user_login, $user_email ) {

if ( empty( $_POST['role'] ) || ! empty( $_POST['role'] ) && trim( $_POST['role'] ) == '' ) {
     $errors->add( 'role_error', __( '<strong>ERROR</strong>: You must include a role.', 'mydomain' ) );
}

return $errors;
}

//3. Finally, save our extra registration user meta.
add_action( 'user_register', 'myplugin_user_register' );
function myplugin_user_register( $user_id ) {

$user_id = wp_update_user( array( 'ID' => $user_id, 'role' => $_POST['role'] ) );
}

and it works as expected on default WP registration page.

The theme I use for my projet has a custom modal login/signup window and I would like to have this code working in this window as well.

See default registration window : https://capricesdelectables.com/wp-login.php?action=register

See custom login box : https://capricesdelectables.com (click on “Connexion” top right red link)

This modal window is templated in a specific mytheme-login.php file

How should I call the code from functions.php to make that displayed in this modal window ?

Thanks