It is illegal to keep the majority of animals native to the United States in captivity in this country, especially birds, without a permit. That is a law that was passed many years ago. I really don't think anyone is going to invade your house and take the bird, though. Or mine, for that matter. If a pet store is selling them, they better be careful. My mourning dove isn't the only technicaly illegal animal I have. I also have a couple of house finches and a a pair of rainbow buntings, but none of them would make it on their own in the wild, so I don't feel too guilty.
Are you sure it is a mourning dove? From what you describe about the tameness, and also a pet store selling them, it almost sounds like it might be a domestic ringneck dove that got away from someone. What does it look like? Ringnecks are very common as pets. Also, I hate to tell you this, but if it coos all the time, it is a male. Female doves are very, very quiet. I have female emerald doves that I had for a couple of years before I ever heard a peep out of them.