Hi,
What kind of licenses Office 365 and 2016 did you have?
As you said, you assigned Office 365 license and Office 2016 license for the same Microsoft Account. I suggest you go tohttps://account.microsoft.com/account and check if you have 2 versions of Office for the Microsoft Account, please take a screenshot here.
Did you install one-time-purchase Office 2016? If yes, I suggest you refer to this support article and check if you can follow these steps to activate Office 2016 in the new PC:
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