My friend and I recently purchased Rainbow Six: Siege on Steam. However I have a very limited data plan so I can't download. Can I copy the game contents from his PC to mine?
Yes you can.
The only check Steam is doing, is that you have Steam running with a logged account possessing the game when you launch a steam game.
So you can totally take the game on the computer of your friend where he installed it and copy it into your Steam games directory.