Hello Matthew, this is Emily.
Considering it was purchased in 2021, the codes you received would be more of a redeem codes vs. license key, and you would use the redeem codes to activate it into your Microsoft account, then moving forward, your Microsoft account is your license as digital license.
Do you still have access to the old computer? If so, you will need to make sure you uninstall it from the old computer. Then on the new computer, go to www.office.com, sign in with your Microsoft account, then click on the upper right "Install Office" to download the installer and run it to install.