Shipping address set to billing address

Commerce shipping is not supported at the moment, so it's probably Klarna copying the billing address by default.

You can do something like this:

yourmodule.services.yml:

services:
  yourmodule.klarna_session_subscriber:
    class: Drupal\yourmodule\EventSubscriber\Commerce\Payment\Klarna\SessionSubscriber
    tags:
      - { name: event_subscriber }

src/EventSubscriber/SessionSubscriber.php:

<?php

namespace Drupal\yourmodule\EventSubscriber\Commerce\Payment\Klarna;

use Drupal\commerce_klarna_payments\Event\RequestEvent;
use Drupal\commerce_klarna_payments\Klarna\Request\Address;
use Drupal\commerce_order\Entity\OrderInterface;
use Drupal\address\AddressInterface;
use Symfony\Component\EventDispatcher\EventSubscriberInterface;

/**
 * Gathers the session data for Klarna order.
 */
final class SessionSubscriber implements EventSubscriberInterface {

  private function getShippingAddress(OrderInterface $order) : AddressInterface {
    // Replace this with your custom fetching logic.
  }

  /**
   * Responds to RequestEvent.
   *
   * @param \Drupal\commerce_klarna_payments\Event\RequestEvent $event
   *   The event to respond to.
   */
  public function onSessionChange(RequestEvent $event) : void {
    $order = $event->getOrder();
    /** @var \Drupal\commerce_klarna_payments\Klarna\Request\Payment\Request $request */
    $request = $event->getRequest();

      try {
        $shippingProfile = $this->getShippingAddress($order);

        $shippingAddress = (new Address())
          ->setCity($shippingProfile->getLocality())
          ->setEmail($order->getEmail())
          ->setGivenName($shippingProfile->getGivenName())
          ->setCountry($shippingProfile->getCountryCode())
          ->setFamilyName($shippingProfile->getFamilyName())
          ->setStreetAddress($shippingProfile->getAddressLine1())
          ->setPostalCode($shippingProfile->getPostalCode());

        $request->setShippingAddress($shippingAddress);
      }
      catch (\InvalidArgumentException $e) {
      }
  }

  /**
   * {@inheritdoc}
   */
  public static function getSubscribedEvents() {
    // Both of these must be run after commerce_klarna_payment module.
    $events['commerce_klarna_payments.session_create'] = ['onSessionChange', -100];
    $events['commerce_klarna_payments.order_create'] = ['onSessionChange', -100];
    return $events;
  }

}

Supporting the shipping address is part of my proposed patch in #3013645-2: Respect the VAT on shipping fee order items

This should be fixed in 2.x.