These are what we call the 'YMMV items.' Things that some people find in this work. We call them 'your mileage might vary' because not everyone sees these things in the same way. This starts discussions in the trope lists, a thing we don't want. Please use the discussion page if you'd like to discuss any of these items.
Which is a real shame because that preceived premise isn't even true! She's neither a "pet" in any sense nor taken by the boy. He and a few of the other characters are essentially her care takers.
Arc Fatigue: Jin's refusal to enter a relationship with Misaki. They do eventually get engaged, but after several, tedious episodes focusing on Misaki making strong advances and Jin breaking her heart. Although, YMMV quite a bit on this.
Internet Backdraft: The appearance of Samgyetang in episode 6 caused a huge uproar among Japanese viewers. While it was different from the light novels—originally it was just intended to be the more simple okayu—it rose beyond They Changed It, Now It Sucks levels, as some Japanese viewers accuse that part of a plot to promote Korean culture in Japanese media. It should be noted the anime changed the entire context of the said scene so the change of dish made sense: in the original it was something Sorata made to Nanami for her sickness, but in the anime, it was prepared by Jin to congratulate the publication of Mashiro's manga and Nanami's first audition.
How does it make any sense if they changed the entire context just so samgyetang, a dish rarely (if ever) prepared by Japanese households could be introduced? Other than a forceful inclusion of something Korean into the anime series, the existence of samgyetang served absolutely no purpose whatsoever. Moreover, in the original LN, students of Suiko sang the Japanese anthem upon graduation, but this was cut from episode 23 of the anime. Instead, the students resorted to singing Suiko's school anthem. Justified or not, some viewers saw this as an overt effort by MBS to appease their Korean investors.