Finally.
I have successfully linked my account!

I tried to figure out what can be the problem and noticed that the SC:FA was marked as private, therefore FAF cannot recognize I have SC:FA regardless my privacy settings.

To help someone in the future:
Navigate here in the steam via browser:

Then search for Supreme Commander and Supreme Commander: Forged Alliance and click to this icon(picture) or click on the three dot and select "unmark as private"

To be honest I don't have any idea why it was marked as private because I haven't marked it before.

GL & HF everybody!