If you see the services that the shop's provide, there's no FS, and they clearly state, that soliciting FS is illegal. However if two adults like each other, they can do what they want. So the girl have to mention it to you, and you give her the donation.
As for the law's, the Japanese were fine with the traditional way of managing prostitution. But McArthur wife was puritan, and she wanted to ban it. So the Japanese made the law's very complicated, to save face. As for the police, they also get a donation.