Something that's adopted has been deliberately chosen. Your adopted country is the place where you choose to live, not necessarily the one in which you're born.
If you describe yourself as adopted, it means that you were taken in and raised by parents who didn't give birth to you. Just as your adoptive parents chose you to be their child, other adopted things are also chosen: an adopted language is one you learn and then choose to speak, and an adopted state is the place you freely decide to live in. The Latin root is adoptare, "choose for oneself."