Processes in Microsoft 365 for setting up Office apps, redeeming product keys, and activating licenses.
When you install and activate Office, that licence should become associated to your Microsoft account online as a digital licence and it is form there that you can reinstall Office without the need for a product key.
Do you find the Office licence on this link and are you able to install Office from there on the other PC.
https://account.microsoft.com/services/
If not, what version of Office did you purchase, is that Office Home 2024, Office Pro Plus 2024 or a Microsoft 365 subscription.