@cookie_honey well they both relate to the feeling of not fitting, and between the flashback and the fire, it seems both have a duty that they feel they must fulfil even at the expense of their own feelings. They aren't making much progress, but there's at least curiosity on both sides, so if there is any romance, it seems it will be a pretty slow burner.
Regarding the king, it was stated chapters ago that the king fears the position of the sun knight. Clearly, they only won the war because of her (as evidenced by them loosing pretty much every fight without her), so most of the people in the kingdom (specially those that weren't affected directly, or benefited) would probably rather follow the sun knight than the king. He seems to think that just by giving her orders and letting other people see that she follows them, that she is a subordinate instead of a threat to the throne, because if that were the case, people would definitelly side with her instead.
It's also the reason why he forces her to remain shrouded in mistery, and keeps sending spies there. Not only her image, but if the fact that the invincible sword of the kingdom is sick, it means she can be killed, and a kingdom without it's trumph card wouldn't be able to fight off a rebellion.