Yes hub protect is per hub, remote admin is across the account.
If your hub had failed and they warranty replaced it then it would have migrated to the new hub.
However if your planning on shutting down your C7 there is a form you can fill in which will migrate the remaining days left on your C7 to your C8. Your then cancel it on the C7.
Your just need to remember to then manually renew the C8 when the time comes.
As you have protect its just a case of following the guide which are plastered all over here on their docs website or main website.
YouTube video from there main website for example.
There are not many vendors who offer extended warranty in the smart home industry. Usually, when a controller dies and the user is out of warranty, the user must buy a new hub and start over. At Hubitat, we know how time consuming is to start over from scratch, so we decided to offer Hub Protect. A unique service that provides a free hub and ways to quickly recover when the unexpected happen. As with most extended warranties, the service is not transferable and non-refundable. However, you may want to check this post for an alternative solution for those who migrated to a new hub and may no longer need the old hub to be protected:
Never had an issue "moving" hub protect to a new hub per Bobby's post. I think Hubitat offering a way to get coverage on the new hub by dropping the old is more than fair. I doubt Samsung or LG or GE would let you do that on a $x,000 appliance with an active extended warranty.