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.
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.