Microsoft 365 features that help users manage their subscriptions, account settings, and billing information.
I do not have to use the original product key from the family subscription purchase? I can just download? Should I wait till my personal subscription expires?
You will have to let your Personal subscription expire before you can accept the invite to share Microsoft 365 Family.
You can't use a product key to install Office using the shared 365 Family subscription. The product key was for the initial install only, or, more specifically, for adding the subscription to the owner's account (the owner = the person who pays for the subscription). The owner will have to invite you as discussed in the article at https://support.microsoft.com/en-us/office/share-your-microsoft-365-family-or-premium-subscription-b389b9ce-3ae3-4a82-9017-39d79972fcba .
After you have received the invite, 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 .