Processes in Microsoft 365 for setting up Office apps, redeeming product keys, and activating licenses.
You are correct in assuming that you need two licenses to install Office Home 2024 on two computers, because a single license will only let you install on one computer (PC or Mac).
I take it you didn't buy directly from Microsoft? Contact the seller and try to find out why you didn't receive the product key for the second purchase.
You can associate more than one perpetual license with the same account and, once that has been done successfully, you install Office using your account at https://account.microsoft.com/services .