Update: I followed the hints below, but the extension Attribute won't be saved. Do you have any suggestions to solve it?

I created an extension_attributes.xml file:

<config>
    <extension_attributes for="Magento\Customer\Api\Data\CustomerInterface">
        <attribute code="user_name" type="string">
           <join reference_table="customer_extension" reference_field="customer_id" join_on_field="entity_id">
              <field>user_name</field>
           </join>            
        </attribute>
    </extension_attributes>
</config>

And an Observer which is registered on the "customer_register_success" event:

<?php
namespace My\Module\Observer;

use Magento\Framework\Event\ObserverInterface;

class AfterCustomerSaveObserver implements ObserverInterface {

    protected $_logger;
    protected $_customerExtensionFactory; 
    protected $_customerRepository;

    public function __construct(
            \Psr\Log\LoggerInterface $logger, //log injection
            \Magento\Customer\Api\CustomerRepositoryInterface $customerRepository,
            \Magento\Customer\Api\Data\CustomerExtensionFactory $customerExtensionFactory
    ) {

        $this->_logger = $logger;
    
        $this->_logger->addDebug('AfterCustomerSaveObserver_Constructor_Begin');

        $this->_customerRepository = $customerRepository;
        $this->_customerExtensionFactory = $customerExtensionFactory;

        
        $this->_logger->addDebug('AfterCustomerSaveObserver_Constructor_End');
    }
    
    public function execute(\Magento\Framework\Event\Observer $observer) {
    
        $this->_logger->addDebug('AfterCustomerSaveObserver_Execute_Begin');

/*      try { */

            $event = $observer->getEvent();
            $customer = $event->getCustomer();  

            $customerExtension = $customer->getExtensionAttributes(); 
            if ($customerExtension === null) { 
                $customerExtension = $this->_customerExtensionFactory->create(); 
            }       
            $customerExtension->setUserName("Test123"); 
            $customer->setExtensionAttributes($customerExtension); 
            
            $this->_customerRepository->save($customer);
            //$customer->save();

/*      } catch (\Exception $e) {
            $this->_logger->critical($e);
        } */
        
        $this->_logger->addDebug('AfterCustomerSaveObserver_Execute_Begin');
    }

}