To be short: What do to if you have a Visual Studio product key, but can't find how to activate it?
In my case this was VS 2017 Professional.
I was triying to find how to activate VS. After few unsuccesfull attempts I've Googled for it and found that there must be a "Unlock with a Product Key" button.
But in my case there wasn't such button. I had absolutely the same window just without "Unlock with a Product Key" option.
After I spent some more time with Google, I found an official MS article. And there was an interesting detail:
Start Visual Studio in Run as Administrator mode.
I tried and it worked. The "Unlock with a Product Key" option appeared instantly.
Sorry if I'm doing something wrong. I wasn't sure if it's okay to answer my own question, but I really dissappointed when I hadn't found any similar questions on SO.