Carrot serves as a foil to Vimes' character. Vimes competely abhors Kings, ridicules the commonly held belief that there's something magical about Kings, believes that Kings are inherrently dangerous because the absolute power they hold. Carrot is a living demonstration that although Vimes should be completely right, he's completely wrong. Everyone loves Carrot not because he's a likeable person, as we see in the later books he's complex and there's some dark currents there, but because he's a real King.
I think Carrot and Vimes are an echo to Rincewind's early conviction that there was a better way than magic. The irony of fantasy characters with a real world mind, they don't want to accept the fantasy world they live in, but they can't escape it.
Or possibly I analyse too much
Contains Strong Language, Fantasy Violence and Mild Peril
The Sunshine Sanctuary says - Click to keep the dragon alive!