Hi Markus,
The Touch ID function on an iPad does indeed work just like that of an iPhone; the button is paired to the motherboard with a proprietary encryption algorithm that only Apple has access to. As a result, only Apple can replace the button and restore Touch ID, but unfortunately, they don't offer repair services for iPads.
So yeah, basically you're out of luck as far as replacing a broken button. You can get one that will function normally as a home button, but say goodbye to Touch ID.
There are some specialized services like iPad Rehab that have the skills to repair damaged buttons, but of course there's no guarantee. That's going to be your best shot at getting it working again, but it's likely not going to be cheap.