Sorry to hear you are having trouble.
For both of you, the activations are happening with your email addresses and keys, and successful, here It leads us to believe that, for some reason, the activation file is not being able to be written to your local PCs:
\ProgramData\MadScientist\The Video Game Machine\license.sig (you can check to see if its there please, likely not).
Perhaps a permissions issue - perhaps disabling AV software for the install?
If that fails, we would like to see an activation log file each of you should have:
\Users\yourWinAccountName\AppData\Local\MadScientist\The Video Game Machine\SASLog.txt.
If you could post them to a cloud service (GoogleDrive, OneDrive, DropBox, PasteBin) with a shared link here, we will take a look at them.
Stardock Customer Service Manager