Hi @François-Olivier Loignon
Please make sure the account which bound with Office key is correct, then I suggest you login to "Services & Subscriptions", check the products you've purchased.
If you have an Office 365 subscription, you can de-activate Office on the old Mac and uninstall Office from the new Mac, re-download the Office suite on your new Mac to install.
If you have a perpetual license, such as Office 2016 for Mac, then you have to reinstall Microsoft Office on your old Mac first.
But according to your description, you changed your hard disk, if you have no way to do this, I would suggest you contact Microsoft to seek help.
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