Magento 2 Save custom customer address attribute in customer edit form using extension attributes

Question

I want to save custom customer address attribute in Customer Edit Form under My Account section. I have created an attribute using Setup script and save that attribute using extension attributes.

Vendor/Module/etc/extension_attributes.xml

<?xml version="1.0"?>

<config xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:noNamespaceSchemaLocation="urn:magento:framework:Api/etc/extension_attributes.xsd">
    <extension_attributes for="Magento\Customer\Api\Data\AddressInterface">
        <attribute code="city_id" type="string"/>
    </extension_attributes>
</config>

Here I couldn't extend Magento\Customer\Api\Data\AddressInterface, that's why I'm using extension_attributes. Then I created a plugin to get and set that attribute value as follows:

Vendor/Module/etc/di.xml

<?xml version="1.0"?>

<config xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:noNamespaceSchemaLocation="urn:magento:framework:ObjectManager/etc/config.xsd">
    <type name="Magento\Customer\Api\AddressRepositoryInterface">
        <plugin name="customer_city_id" type="Vendor\Module\Plugin\CustomerCity"/>
    </type>
</config>

Vendor/Module/Plugin/CustomerCity.php

<?php
namespace Vendor\Module\Plugin;

use Magento\Customer\Api\Data\AddressInterface;

class CustomerCity {

    public function afterGetById(
        \Magento\Customer\Api\AddressRepositoryInterface $subject,
        $address,
        $addressId
    ) {
        try {
            if ($address->getExtensionAttributes() === null) {
                $address->setExtensionAttributes($this->extensionAttributeFactory->create(AddressInterface::class));
            }

            $extensionAttribute = $address->getExtensionAttributes();
            $extensionAttribute->setCityId(
                ($address->getCustomAttribute('city_id')) ?
                $address->getCustomAttribute('city_id')->getValue() : ""
            );
            $address->setExtensionAttributes($extensionAttribute);
        }
        catch (\Exception $e) {
            // Don't handle exceptions
        }
        return $address;
    }
}

Unfortunately, I'm getting an error We can't save the address. As I printed an exception message, I'm getting this message: Class string does not exist.

Please anyone let me know what I'm doing wrong. Thanks in anticipation.