If a commercial customer has an item delivered to another company in Austria and the invoice is sent to Germany, then the corresponding VAT ID should be taken from the delivery address - and not from the billing address.
The delivery address is the basis for which tax rate is used for the order.

The VAT ID. field is not available under “Change delivery address”.

Registered customers change the VAT ID in the profile settings and cannot change it at all via the billing address, as guest customers do it.

The behavior should be the same for both account types and ideally you should also be able to store multiple VAT IDs.