Recently we upgraded our testing environment from Magento 2.1.2 to 2.1.3. Soon after the upgrade we noticed a lot of functions in our multi-store setup were not working properly anymore. A lot of the configurations in the backend were messed up, but the worst were the store configuration itself and the theme configuration which both just went back to the default settings on every save.
Hopefully this is fixed in Magento 2.1.4, untill they fix it there is an easy workaround I found on GitHub. It involves making a minor change to a Magento core file, but it should get overwritten on the next update.
In your Magento Root dir navigate to vendor/magento/module-store/Model/Config/Processor/Fallback.php
On line 171 you should find:
$code = $website['website_id'];
Change that line to:
$code = $website['code'];
There you go, your multi store setup should work again. Make sure your .htaccess is still properly configured for your multistore setup after the upgrade.