“iPhone required” error on child’s Apple Watch hearing aid app

The app developer would need to write it so that it will work on a Watch without a companion phone, if possible. It's not possible for all features to work under these circumstances, though.

I suggest contacting Oticon and discussing the matter with them. Your audiologist may also be helpful.

It has nothing to do with Apple IDs.

When you have a family members Apple Watch paired to your iPhone, certain apps require an iPhone to use. This ensures that both devices are using the same Apple ID and apps and services can sync and communicate between the two devices. Unfortunately, that app wont be able to work on your child’s Apple Watch, you can try looking for other ones without the iPhone required.