The Prince knew that Corin would say something like that. People who were not royalty, people who had never been to court or knew how this world worked, had a difficult time understanding why they could not just do as they pleased.
"No," Sora replied firmly. "Riku has never been a secret to anyone who has bothered to pay attention." What Corin didn't know is that Sora had gotten by under the radar because people didn't pay attention to the Prince in general, let alone who he was with. "We are not officially engaged." Sora was only barely officially courting him. So if Corin told that to anyone, he'd be lying.
"I promised Riku that I would protect him no matter what," and if that meant removing his brother because he was going around saying things he shouldn't, Sora was going to do that. "Corin, as soon as people become aware that you are his brother, the questions will not stop." People will be constantly bugging him for info about Riku, their family, his past. "My world is very different than yours, and Riku has not fully crossed into it yet. If you speak too much on his past, if you put a stain on his reputation, he will suffer." Did he get that? Sora didn't like it any more than Corin did, but that was the way things worked.