This seems like a lot of thought was into it already, but at the same time, it is rendered unplayable by some really obvious problems...
You cannot tell what preferences a "pet" has before having to decide to keep it forever or sell it, which means selling makes sense exactly when you can't handle any more pets, but simply can't turn off trapping.
You also can't rename pets, so while giving them a name as an identifier does make it slightly less confusing, the names can't have any meaning.
Pets can have the same "like" and "hate", which makes zero sense and ends up happening more often than you would expect and basically prevents those pets from ever having any obedience, whatever obedience does.
Assignments have to be redone every single day, which becomes extremely tiring after like 5 days. Why is there even a "confirm" button?
All of these are really obvious and trivial problems with massive impact. I just don't understand how something like this can happen. I feel bad for criticizing the game so hard, but I really feel like all of these problems could be fixed in like 5 minutes when everything else in this game probably took days to make. I just don't get it...