custom error message for empty username and password using authenticate filter not working – Woocommerce

The following code is the reason why it doesn't overriding.

if ( empty( $username ) || empty( $password ) ) {
        if ( is_wp_error( $user ) )
            return $user;
        ...
 

Try to remove this condition.

if (is_wp_error( $user ) )
  return $user;

It should be something like this.

/**
Purposed: Custom Login Error Message
Description: This function override the default error message on login form.
**/
remove_filter( 'authenticate', 'wp_authenticate_username_password' );
add_filter( 'authenticate', 'custom_authenticate_username_password', 30, 3 );
/**
 * Remove Wordpress filer and write our own with changed error text.
 */
function custom_authenticate_username_password( $user, $username, $password ) {
    if (is_a($user, 'WP_User')){
        return $user;
    }
    
    if (empty($username) || empty($password)) {
        $error = new WP_Error();
        if (empty($username )){
            $error->add('empty_email', __('The username or email field is empty.'));
        }

        if (empty($password)){
            $error->add('empty_password', __( 'The password field is empty' ));
        }
        return $error;
    }

    $user = get_user_by( 'login', $username );
    if (!$user){
        return new WP_Error( 'invalid_username', sprintf( __( 'Invalid username or email address.' ), wp_lostpassword_url()));
    }
    $user = apply_filters( 'wp_authenticate_user', $user, $password );
    if (is_wp_error($user)){
        return $user;
    }
    if (!wp_check_password( $password, $user->user_pass, $user->ID )){
        return new WP_Error( 'incorrect_password', sprintf( __( 'The password you've entered is incorrect.' ),
        $username, wp_lostpassword_url() ) );
    }
    return $user;
}

Notice that I added braces to every condition to make it more clear.

You might also want to override the function wp_authenticate_email_password.

remove_filter( 'authenticate', 'wp_authenticate_email_password' );
add_filter( 'authenticate', 'custom_authenticate_email_password', 31, 3 );
function custom_authenticate_email_password( $user, $email, $password ) {
    if ($user instanceof WP_User) {
        return $user;
    }

    if (empty($email) || empty($password)) {
        $error = new WP_Error();
        if ( empty( $email ) ) {
            // Uses 'empty_username' for back-compat with wp_signon().
            $error->add( 'empty_username', __( 'The username or email field is empty.' ) );
        }
        if ( empty( $password ) ) {
            $error->add( 'empty_password', __( 'The password field is empty.' ) );
        }
        return $error;
    }

    if (!is_email($email)) {
        return $user;
    }

    $user = get_user_by('email', $email);
    if (!$user) {
        return new WP_Error('invalid_email',__( 'Invalid username or email address.' ));
    }

    /** This filter is documented in wp-includes/user.php */
    $user = apply_filters( 'wp_authenticate_user', $user, $password );
    if (is_wp_error($user)){
        return $user;
    }

    if (!wp_check_password( $password, $user->user_pass, $user->ID)){
        return new WP_Error('incorrect_password',sprintf( __( 'The password you've entered is incorrect.' ),
        $email, wp_lostpassword_url() ) );
    }

    return $user;
}

There are two functions under authenticate, wp_authenticate_username_password and wp_authenticate_email_password. Try to override both of them, maybe one of them has the error message that is not overriding, use the codes above and examine.