It's a good question. I think Vimes knows he is, but does Carrot ever really find out that he is? I think part of the joke is that we all know that Carrot is the true king (how funny to be talking about this on May 26!), but he remains blissfully unaware of it. He acts like the perfect king (of a constitutional democracy), and people react to him as if he were a king, but none of them know why.