You can do just the last two digits for Chinese if it's already clear which century it is (which it usually is, to be fair)

Yes, same here! Though it's been a bit less common since the turn of the century, maybe because we're still used to the two-digit forms of the current years standing for 19-something? Idk.

but the digit-by-digit "format" never change

That's fascinating! Yep, very fun difference - here the two-digit form is always read as a number.