Something smells fishy here, but...
Every version of Mac Office I've ever used (Since Mac Office 4) has asked for a key when any of the apps was opened for the first time. This was even true when multiple licenses were obtained and the package was installed on office computers on a LAN.
Office Mac 2004 and 2008 are interesting because unlike other versions, they check their local LAN for other users using the same key. If you have a single copy and and have multiple users utilizing it (with the same registration key) on a LAN, when a second user attempts to open an Office App, you will be told that another user is using a MS Product with that Identification key and that you can't open any Office Apps. When the other user closes their MS App, you can then open it.
In the case of the OP, it sounds like the product key from the purchased version may have been stolen or otherwise compromised and that an MS product using that key is being used on the LAN. I find it odd that it never asked for a new key though.
This is a case where I would recommend a 3rd party app uninstaller like App zapper to get all the bits of office off the OPs computer and then reinstall from the disc.
If this persists, return the Office package and buy from a more reputable source.