[SOLVED]
I can confirm that fixing this registry keys has worked for me too and fixed the issue.
I've changed the values and rebooted.
HKEY_LOCAL_MACHINE\SYSTEM\CurrentControlSet\Services\sppsvc<key>:

Next time I opened OneNote 2016 on my Win10 it opened properly without the "Enter your product key" message.
Reason
I agree with TimothyMarchini in his post here:
that the reason is unclear, why this issue occurred.
This seems to be another good page describing a fix by fixing the permissions for the service (not tested):
Background
I tried to find the documentation for this keys, which are configuring the services, but it seems that they are only documented explicit in the book:
You may be able to apply the changes by the services.msc user-interface too, because the registry keys and values used, are represented in the user-interface too:
HKEY_LOCAL_MACHINE\SYSTEM\CurrentControlSet\Services\sppsvc<key>