Microsoft 365 features that help users manage their subscriptions, account settings, and billing information.
No product key can be used for second or subsequent installations of Office. The key, if you had one, was only for the initial setup.
Did you invite your family member to share your subscription? See https://support.microsoft.com/en-us/office/share-your-microsoft-365-family-or-premium-subscription-b389b9ce-3ae3-4a82-9017-39d79972fcba.
And did your family member then follow the steps at https://support.microsoft.com/en-us/office/someone-invited-you-to-share-their-microsoft-365-subscription-6a6c098e-1934-4436-9a94-69c92590da5b to accept the invite and install?