You don't need the product key to install MS 365 on another computer - simply follow https://support.microsoft.com/en-us/office/download-install-or-reinstall-microsoft-365-or-office-2024-on-a-pc-or-mac-4414eaaf-0478-48be-9c42-23adc4716658
If you already redeemed a product key, reinstalling the apps, or you have an assigned license through work or school, you can skip ahead and go to the section sign in to download and install Microsoft 365 or Office.
As per https://learn.microsoft.com/en-us/answers/questions/5204492/how-to-add-another-pc-to-microsoft-365
"With Microsoft 365, you can install Office on all your devices and sign in to Office on five devices at the same time. This includes any combination of PCs, Macs, tablets, and phones."
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hth
Marcin