Just like a machinegun, it is the lower that is registered as the SBR. You can own any upper you want if you don't have any AR lowers. You can register any lower you want as an SBR as long as you pay the transfer tax, as the 1986 machinegun ban didn't have any effect on our ability to make new SBR's.
However, I have hear rumor (unconfirmed) that the BATF has ruled that since when you register a lower as a new SBR, you are effectively 'manufacturing' a new firearm, and therefore it doesn't matter if your receiver is pre-ban or not, the effective date of manufacture is the date they approve your tax stamp. Therefore, since the SBR is a semi-auto assault rifle, it is now officially 'post-ban' and subject to all the 1994 slime bill silliness. As I said, that is unconfirmed- check with your friendly local BATF office to make sure. HTH.