Where is the iCloud data on a third-party iOS app stored?
The ~/Library/Mobile Documents/ folder only contains data that is stored in your iCloud Drive. 

iCloud is an umbrella marketing name that also covers other services that are not implemented in the same manner as iCloud Drive. Apps that store data in iCloud directly often do so via iCloud CloudKit. iCloud CloudKit is a service provided by Apple that makes it easier for app developers to store and sync data. As the lower tiers are provided for free by Apple, it is a relatively popular service.

When apps store data in iCloud via CloudKit it is not generically accessible to the user. The app itself needs to provide an interface for accessing the data before you as the user can access the data.

If no such interface exists within the app, you have to contact the app developer to request such an interface.

If you're a citizen of the European Union, you might have luck sending the app developer a GDPR request for your data. That is often a one-time, manual process however.