Moan away. I’d get in touch with your client and let them know that there would be an additional charge for the database if they stay with this host. Or that you could find them a better host for less per month that would include the database.
Yup what they are doing is perfectly legal. They aren’t selling you the MySQL software for a charge or anything, what they are selling is the ability to use that software as part of the service they’re selling. Nothing illegal about that.
