Processes in Microsoft 365 for setting up Office apps, redeeming product keys, and activating licenses.
If you are the person who pays for the 365 Family (or 365 Premium) subscription and you plan to install on a second computer for yourself, just go ahead and install from your account at https://account.microsoft.com/services . For subscription versions, you can always install on multiple devices.
If you want to invite a family member to share the subscription; see https://support.microsoft.com/en-us/accounts-billing/subscriptions/share-your-microsoft-365-subscription. The family member should then follow the steps at https://support.microsoft.com/en-us/accounts-billing/subscriptions/someone-invited-you-to-share-their-microsoft-365-subscription to accept the invite and install Office.