The C drive can be recognised as changed, Windows used to use it as part of the check that it hadn't been installed on a different computer so changing the drive meant re-authorising Windows. I think, though I'm not sure, that the key component for more recent Windows versions is the motherboard - as far as Windows is concerned a new motherboard = a new PC.
Even if Windows doesn't reference the initial installation drive other software might, so stuff might need re-authorising or even reinstalling if the C drive is changed.
There's also the question of whether the MBR, boot sectors and the hidden partition Windows creates nowadays are in some way dependant on the C drive being the original - e.g. the Windows boot system relies on sector based partition sizes and locations to work correctly.
Personally I'd put in the new C drive then do a fresh installation to it and only reconnect the old C drive after that, having made sure BIOS sees the new drive as the boot one. Then copy any data wanted off the old C to somewhere else and reformat the old C to use as a data drive.