AOL "PRIVATE WiFi" is software that is intended to protect you when you connect to public wifi networks. You apparently can "authorize" your home wifi network so that you can connect to it without problems, but that authorization seems to be tied to the hardware. Thus, now that you have a new hardware device, you have to re-authorize things.
What is PRIVATE WiFi™?
PRIVATE WiFi is a virtual private network (VPN) software that protects your identity and sensitive information by encrypting everything you send and receive over public WiFi networks so that your online activity is invisible to threats. For more information, please visit privatewifi.aol.com.
If you go to MyBenefits.aol.com you should be able to sign on using your AOL username or AOL email address and your AOL password.
I have no idea how to "download a different PRIVATE WiFi license" for your new device. Perhaps one of the following links will help. If not, "please visit privatewifi.com/contact and fill out our Contact Form. You will receive a response within 12 hours." If you don't want to wait that long, try AOL's Chat support here --> https://help.aol.com/contact/
https://help.aol.com/articles/how-do-i-deactivate-a-license-for-an-aol-plan-benefit
https://help.aol.com/articles/loading-activating-and-deactivating-private-wifi
https://help.aol.com/articles/installing-uninstalling-private-wifi