That red button in the corner of the game is something we can't change from appearing and we can't change the description. There is no online activation required. We've worked with the devs but it's not something we can change as far as I know.
Even if you attempt to activate and it states it's in demo mode, it's not. It works straight out of the box.
There are some multiplayer keys which are available from support as well and I can see we've missed that as well on the support page.
Sorry for the mix-up, I had hoped the team had added something as the support FAQ for the game but I can see it's not there ;-\
We'll get something added ASAP.

To clear the whole situation:
Our version of Earth 2160 doesn't require activation to play. That's for a start.
The button you can see inside a game is only for Earth.net multiplayer purpose. Since Earth's servers register users by serial numbers they must be informed that your serial is unique - therefore you activate this key in order to play on Earth.net.
To play local/IP version of multiplayer you just need to obtain your own serial code from our support team. And here you don't need to activate your game, you're playing on your own.
And singleplayer is free of any kind serial/activation stuff (which is usual here on gog.com :)
One thing that might be rather stressful is that unlucky text saying that if you do not activate the game you will be playing demo mode - well, this is sth we couldn't change, but believe me - this means nothing - there is no demo mode in our game :)
Best
Grah

Yes, Multiplayer mode requires unique cdkey, if you got one (from our support) then just enter it and LAN as well as Internet play will be available for you :)
Best

If you haven't done so already, please contact support with more details.