Earlier quoted context omitted.
Nope, a Microsoft account is not enough. It must be an @outlook.com address, or any registered company/school/university address. It took me almost 3 days to find the problem. Microsoft changed that and between all "answers" there is only one single thread in the Microsoft forums that had the solution.
What does "any registered company/school/university address" mean? Some years ago, I activated some Office licenses using my company email; we never did any hosting with O365 or whatever was it's predecessor, and at the time, everything went fine. All I had to do was to create live account using that email address.
Which means, that you need to associate your existing account with an @outlook.com address. It seems, that Microsoft changed that requirement somewhere in 2020/2021.
Yes, previously Microsoft account with whatever email address was enough. But they changed that.
I stumbled upon that while upgrading to new hardware, which requires new activation of the Office products.