WordPress' update user email confirmation.

new_email.php

<?php
/**
 * WP Core logic to change user email.
 *
 * This method is monolithic because it depends on admin ui.
 * So if you need custom user page,
 * you need re-implement these features.
 * 
 * @see https://github.com/WordPress/WordPress/blob/master/wp-admin/user-edit.php#L100-L116
 * @see https://github.com/WordPress/WordPress/blob/master/wp-includes/user.php#L2994-L3102
 */

// If a user changes their email address,
// 1. WordPress store new one as `_new_email` usermeta
// 2. Send confirmation email with link trailing hashed email.
// 3. The user clicks email and user_email will be updated.

/**
 * Check if hash is appropriate for user.
 *
 * @return bool
 */
function my_confirm_user_hash( $hash, $user_id ) {
	$new_email = get_user_meta( $user_id, '_new_email', true );
	if ( ! $new_email ) {
		return false;
	}
	return $hash_equals( $new_email['hash'], $hash );
}

/**
 * Customize confirmation mail.
 *
 * @param string   $email_text Some string should be included.
 * @param string[] $new_email  Array with [ 'hash', 'email' ]
 * @return string
 */
add_filter( 'new_user_email_content', function( $email_text, $new_user_email ){
	// Here, you can customize email string.
	return $email_text;
}, 10, 2 );