Yes, Keil uVision requires a licence. There used to be a free licence with limitations as well, check the Keil websites.
> - ARM_PRODUCT_DEF: unset
> - ARM_PRODUCT_PATH: unset
> - ARM_TOOL_VARIANT: unset
This sounds like your installation is not correct, or failed for some reason.
I suggest to consult the Keil support and/or the Keil forum, they are in a much better position to help you.