That is almost always an over-aggresive Malware app so check its logs.
This is a work computer and my organization uses WatchGuard EDPR which is a pretty extensive malware product, so I wouldn't be surprised if it could be causing issues with Stardock software. I only have access to the basic malware scanning log for on my machine, though, so can't really tell definitively if it is the cause. There are no entries in the log that mention any file ending in .sig, anything with the keyword "Stardock" or "Multiplicity", or anything in the C:\ProgramData\Stardock\ path.
FYI - for your own troubleshooting: there is also no real consistency in terms of when it happens, except for two things:
1) When it does happen, it is always after a reboot of the system. I did suspect that Windows Update might be the cause because there are some stretches where this workstation might be running for a couple of weeks straight before it has to be rebooted because of an update, but I've been through a few "Update >> Reboot" cycles more recently where it didn't cause the license to disappear, and the most recent case was a regular shut-down and the re-started the next day (ie, no updates were applied).
2) All Stardock products appear to lose their licenses at the same time (on this machine, I run Groupy, Start11, Fences, and Multiplicity), so whatever is causing the licenses to deactivate, it is doing it to all of the licenses, not just one product.
Also, I did follow the path that you provided to get my MP key, and it worked. Originally I checked my "Orders" page and the MP 4 key is not in there, so that might have something to do with the activation not working properly when I use my credentials.
I would propose at least the following two enhancements to the software to help mitigate this kind of issue:
1) If a product shows no valid license as being active, have it ping your activation server to check for a valid license before disabling itself or prompting the user to re-activate. As part of this, we should be able to save our Stardock credentials securely in the system so that this activation request can happen transparently in the background and not interrupt our workflow.
2) Update your licensing database and your activation server such that it doesn't get confused when someone uses their Stardock account to re-activate their license. The system should be smart enough to see that if there are two valid licenses for different service levels for the same product on a person's account (ie, Multiplicity Basic via an OD subscription and Multiplicity Pro via a direct purchase or upgrade) that it should know to return the license for the more full-featured service level when queried by the software.