To unlock the Honda Civic in the Honda championship, you have to go to:
http://ww1.honda.co.uk/racedriver/
And then you can unlock it.
But, to do this, you need your installation code, which of course you don't need with the GOG installer.
So is it not possible to get the car?
It doesn't really bother me, I just wanted to see what it was like.

You can temporarily unlock it using a trainer.
But as you might expect, the 2006 Civics look and handle about the same as the 2005 Civics, so you aren't missing out on a whole lot.