Processes in Microsoft 365 for setting up Office apps, redeeming product keys, and activating licenses.
The key detail is whether you purchased a Microsoft 365 subscription or a one-time Office license. Microsoft 365 allows use on multiple devices, while a one-time Office purchase covers one computer.
Here are some suggestions you can try:
Open Services & subscriptions and sign in with the account used for your purchase. Check the exact product listed. If it is a one-time purchase already used on the new Mac, that same license cannot cover both computers.
If you have an active Microsoft 365 subscription, sign out of Office on the older Mac, then sign back in with the account that owns the subscription.
If activation still fails, close all Office apps on the older Mac and run the Office License Removal Tool. Have your account password ready, since it removes all Office licenses on that Mac. Reopen Word, sign in again, then restart after activation.
If it still says “denied,” please share the complete error message, product name, and older Mac’s macOS version. Hide your email address and any product key.