It’s really easy for any king to meet someone new and instantly place that person in a category: “Hooray! Here’s another subject!”
Indeed, this king sees subjects everywhere. However, his authority has a twist:
“For what the king fundamentally insisted upon was that his authority should be respected. He tolerated no disobedience. He was an absolute monarch. But, because he was a very good man, he made his orders reasonable.” (10.15)
By trying to be “reasonable,” the king ends up having no real power. He only gives orders that will be followed anyway. For instance, when the prince asks him to order a sunset, the king says he will order the sun to set when “conditions are favorable” (10.34)—that is, he will order the sun to set at the time when it will set in any case.
The king also doesn’t actually have any subjects. He’s the only person on his planet. Without the prince there, his only other potential subject is a rat. So, while the king claims to have power and absolute authority, he’s really all alone with nothing to rule over.
From an allegorical standpoint, the king symbolizes rulers who make a big deal about the power they have, but who in actuality are pretty ineffective at enforcing their power. It also mocks their grandiosity and showiness, which is kind of funny because they think they are way more important than they actually are. The king is somebody who thinks he’s boss, but he really isn’t.