If you go to GM and ask for "transaxle" oil, you're liable to get something to put in front-drive transaxles.
People say a C5/C6 has a "transaxle" but that term is correctly applied to a transmission and drive axle system that is one unit, such as you find on may front wheel drive cars where the transmission and the drive axle are in the same housing and share the same lubrication system.
C5/C6 have the transmission and drive axle bolted together, but they do not share the same housing nor do they share the same lubricant. In spite of that, the assembly is commonly referred to as a "transaxle".
For your C6 you need to purchase GM's rear axle lubricant it sells for Corvette drive axles and you need to purchase a different lube for the transmission, either Dexron III ATF for both automatic or manual transmissions.