An App Service is literally just a VM running IIS. Everything is the same provided you configured the app service to actually run .NET which is an option when you set up the service. IIS is running in the background, your site is deployed to a specific folder and IIS is adjusted to host it just like on a normal IIS instance.
My gut instinct is that your configuration isn't correct. However I should also point out that app services are configurable via the slot settings that your site is hosted under so it is also possible that you have overridden the configuration settings. Go to the Azure portal and then to your app service. Under Settings\Configuration go to the Default Documents tab. This is going to override your config settings I believe but it is possible it might pull from your config to get the list. I've never bothered to look.
Use Kudu to view your site's file system and ensure the web.config is properly deployed. Under the app service in the portal go to Development Tools\Advanced Tools. Click Go to bring up Kudu. Personally I then use the nav at the top to go to Debug console\Powershell. You can see the file system and make changes if needed.