Welcome to the Microsoft Q&A (Preview) platform. Happy to answer your questions.
When you move the VM to another resource group, the VNET is not changed. Your VM still connected to that VNET where it is initially deployed.
Resource Group is just a container where you can deploy resources. Those resources can be in the same region or may be of different region. But in your case when you tried to change the VM to a different Resource Group, your VM is still linked to the same VNET and hence you are not allowed to attach NIC which is from a different VNET.
As you mentioned, the only way is to delete the VM and create a new VM in different VNET in different RG as you desired.