If she was the kings property or pet the king wouldn't have called her his daughter when William tried to kill her.
And at the title awarding ceremony, he wouldn't have treated her that same way an overaffectionate father would.
However we all have our different opinions. So let's just stick to what we believe.
I think its stated pretty clearly that a daughter means less than human in that world. Plus, i think people wont be happy if someone try to kill their property. What im saying is that their love isnt condition and is not of equal. They love her as a pet because she was loyally sucking up to them and of use to pleasing them. She was an interesting toy that entertained them because she did whatever they asked and made them feel special. Even till now, the oldest brother has an unfavoured wife that he shuns, he also uses his magic just like his brother to force women int kowtowing to him. The father has a ton of women in his harem that he abuses and daughters he executed. None of that have been rectified. And yet author writes it as if that's cute. Its clear in the Raws. When they found out that she was thinking of bringing equality, they did a 180 and put her on trial.

Can we all just appreciate how the princes and the king literally got upset(or pissed) at how Chris said 'my girl'?
For people who grew up in a male dominate world, the king and princes have had a major character and personality developement. They went from treating the princess like crap to protecting every hair, literally every hair, on her head.
Knight Lee included. He called her 'our princess' when William tried to kill her again.
Though the king still has a ways to go until he realizes Jinsu is taking his precious daughter.