Slitherine

I just purchase Gary Grigbys War in the East through Steam. I saw a serial number in a dialog box during the purchase and copied it. Later I did get prompted to enter it when I was logging in.

Now my friend said that to play multiplayer, I have to register it on the matrix site but can't find the serial number?

Is it in a file in my game somewhere?

You don't have to register the game on a Slitherine/Matrix account if you have purchased the game on the Steam store.
Just login with your account details.