It also appears that you're using a student version of Office, the licensing terms are most likely different for that version.
Even Apple, the poster child for the walled garden, locked down app stores we all love to hate doesn't limit how many devices you can have a given application installed on. They do limit the amount of devices that can protected music and videos can be played on, but even then they give you a once per year "deauthorize all" button (with exceptions granted with a quick call to support) and a de/reauthorize button in the app!
"Depending on your needs, you might find a single outright copy of Office Home & Student 2013 is more economical than an Office 365 subscription -- but not if you can't keep using it after you next upgrade your computer."