Strictly speaking, you can only legally buy a game for one computer. This can cause problems with upgrades where all you have in common with the originally computer is the keyboard.
You are allowed to make back up copies to protect your investment but not for sale or hire.
If you wish to run it on two computers, their connection to the internet is probably what is telling them they have the same game. Some games wont let you play if it is not registered via the internet on each installation to prevent people disconnecting the internet and playing anyway.
If you wish to play LAN with another person, legally you should own two copies. However, some games will allow you to do that. There are also illegal "cracked key" downloads that will enable you to play on two computers, but if you use the internet with them you run the risk of having your genuine copy licence revoked, your internet provider asking you to please explain why you should not be disconnected and legal action from the publishers.
Having said all that, I caution you about going to websites that have a "cracked key" to enable games to be played on LAN. They are sometimes sponsored by porno web sites and can be riddled with viruses.
The problem is simple: it costs millions of dollars usually to develop and market and game. Most of the profit is in publishing and distributing with the developer having the majority of the expense, the majority of the risk and the minority share of profits. Some games are no longer available because they were economically killed by piracy.
If you wish to play a LAN game with another member of the family, disconnect the internet and see if it will let you. I would be very surprised if anyone was ever prosecuted for this.