It is the situation when I am logged into WordPress website as a admin user and change the email id of any other user.

As soon as I change the email id of any registered user from the admin user section, then automatically a mail sent to the user
with below mentioned content: 

Hi [username],

This notice confirms that your email was changed on [website].

If you did not change your email, please contact the Site Administrator at [admin email] 

This email has been sent to [user email] 

Regards, 
All at [website] 
[website url]

This is a feature since WordPress version 4.3.0 to improve security measures.

In some cases we want to modify the email content to make it according to us in spite of using as default. So to in this case
we need to use a WordPress predefined filter that is:

email_change_email

It filters the contents of the email sent when the user’s email is changed. Check in detail here: https://developer.wordpress.org/reference/hooks/email_change_email

So, we need to use this filter to modify the email content sent when we change email id of any user. Just copy and paste below mentioned code in
functions.php file of your currently activated theme: 

 
    function custom_email_content( $email_change, $user, $userdata ) {

        $modified_email_message = __( 'Write your modified content' ); //put your modified content in this section

        $email_change[ 'message' ] = $modified_email_message;

        return $email_change;

    }
    add_filter( 'email_change_email', 'custom_email_content', 10, 3 );

 
Just put your modified content for the email as per your required format. Check by changing any user email id from admin, automatically a mail sent to the user with your modified content.

How to stop automatic mail to user on email change

Sometimes you need to stop this feature of automatically sending email to respective user when you are changing email from admin.
In that case just paste this below code to your functions.php file of your currently activated theme.

add_filter( 'send_email_change_email', '__return_false' );

It will stop sending mails to the user when you change their emails from admin. “send_email_change_email” is a filter which filters whether to send the email change email. It is part of the wp_update_user() function.

It is advisable to avoid stopping these emails because it a part of security. Just modify the email content as per your needs will be a better choice.

Always customize WordPress with proper WordPress way 🙂