The reason for those 3 installations from microsoft is so that you can upgrade your computer throughout it's age (changing various components alters that number that was mentioned earlier, so reinstalling the software would count as another install) without allowing you to just install it on as many computers as you like -

People who reinstall the software multiple times on the same computer can usually just call microsoft to unlock the license if it's used more than 3 times - it's a roadblock of inconvenience more than anything.

You shouldn't have a problem installing the operating system on both computers, just be aware that you aren't using it 'correctly' - a single upgrade of 7 is supposed to be used on only one computer.

Windows activation is a remarkably complicated system, as they want to allow someone to reinstall the same computer multiple times, but not allow someone to install the same copy of Windows on multiple computers. The way they do this is they get several somewhat unique numbers to your system (CPU serial number, network adapter's MAC address, and so on), then mathematically combine them into a number that's unique to your computer, but doesn't mean anything else. Then they send that number and the Windows CD key to Microsoft's magic master servers, which look up that key, and make sure it hasn't already been used, or that it has been used for the same computer only. The server sends the OK, the computer finishes the install, and everything works.

Obviously, it's rather straightforward for Microsoft to set up CD keys that will work for a given number of computers other than one, or CD keys that will work for any number of computers but make a note to a central account somewhere, or so on.

My one concern would be whether the disk is good for an install and then two reinstalls, or whether it's good for running three systems concurrently. That would definitely be something worth checking up on before you install it on the second box.