Hi Thomas,
You Creative Cloud membership does give you the entitlement to install the software on two machines.In case you are being prompted to enter a license key when launching the software on a second PC, it is likely that you may not be signed in with the ID you have you Creative Cloud membership under. I would advice you to check the ID on the machine where you are able to use the applications and on the machine where you aren't to make sure they are both the same.
In order to check the ID you are logged in with, please open the Creative Cloud desktop application, click on the settings icon and choose preferences.
Thanks.