One of the biggest problems would be enforcement of that license. With driving, cops are everywhere and regularly pull people over to check their license. With hunting, there are game wardens that patrol hunting areas and check the licenses of hunters.

With procreation, people can have unprotected sex anywhere and typically in private. You’d either need to give some group of people permanent access to enter any private space at any time (to randomly check for unlicensed sex), or force everyone without a license to take birth control or be sterilized. Unfortunately, none of those options are ethical.