TL;DR: If you bought a Forged Alliance copy on GOG you can not play on FAF as of now. We are trying to work this out with GOG.
Supreme Commander: Forged Alliance is now available on GOG! Usually this should be a reason for celebration. Unfortunately it’s not. As you may have noticed our Steam Link Mandatory hint was not changed on the registration page. And that’s not because we are lazy or missed it, but because it remains still valid.
GOG contacted us roughly two months ago and discussed option to support GOG on FAF. We told them about the grey area legal situation around FAF and the reason for the Steam Link requirement. Our proposal was to add a feature so that we add GOG Link in addition to Steam Link. However, GOG wanted to go further and remove our requirements for any account linking altogether. Therefore they contacted Square Enix but got no sufficient response so far.
As GOG has no online service themselves, GOG users currently have no option to play online. We are sorry about the inconvenience, but currently it is not in our hands to change this situation.
— Brutus5000