You're plenty fine to install the game on as many machines as you want to -- goodness, if you want to play on your laptop and main machine we sure don't care (and in fact are glad you want to play that much).
In terms of multiplayer, each player connected to a server will have to have a unique license key or it won't let the second player on. This is done for a variety of reasons, among them authentication -- you won't have to "log in" to a server, but the server will be able to identify players and admins and so on by their license key prefixes, which act as a unique identifier. There's no central server or tracking or any of that mess, so it's all just negotiated between the players and their own server.
