Yes,
not good news.... I've also confirmed this - and the challenge response mode for older versions/software such as Dimension Pro, Z3TA+ (version 1), Sonar X1/2/3, Rapture (not pro) - and I'm sure others as well - are all tied to the specific computer using some form of key based on the hardware (and potentially software, etc.).
As a second test, I generated the request key on a different machine (testing with Rapture Pro), and the requests (which you can open up in Notepad to see - they are simply an encrypted text string) are indeed different

.
One thing that I can't easily test (without regressing the BIOS and risking all that), but which I'm pretty sure about: I've not noticed any Cakewalk software being like some other programs which need a new key every time you have to update the machine's BIOS! I just hope that this is the case...
Anyway, I'm going to try to play with ways to create a set of keys for my Music PC - at least making this resilient to future issues - but I suspect it will need me to create a new Windows bootable disk, install on that and then create the keys (hoping that the key is not also tied to anything specific around the hard disk drive). But that's for another day when I have a bit more time!