If one house is "the place where I get attention from dad and I get to pick the games/movies/series we play/watch, and I can be at ease" and the other place is "where I'm forced to do chores the whole time for a bunch of kids I don't care about, and nobody listens to me and I just have to keep my head down", of course she's going to feel one's better than the other.
Mind you, that might as well be one house "the place where i'm stuck all alone with my dad and there's nothing interesting to do and he's always busy and I just wait around" vs "the place that's always full of life with a bunch of kids to play with, lots of interactions, always stuff going on and new things to do". It's not necessarily "being a single child" that is more or less fun than "being in a group" (I never got the impression any of
@strawman's kids didn't feel loved and at home - whhile my niece, who's an only child, is pretty vocal in how she doesn't fel appreciated because neither of her parents actually spends time with her) . It's also partially down to personality, and how authority and tasks and time management are handled on both locations.
Parenting, especially co-parenting, isn't a competition, or at least it shouldn't be. In the best of cases - which yours is not :-P - it can be a cooperation and everybody can help develop talents and give the kids homes they enjoy. In the worst case, it's a constant competition for being the "best" parent and the "good guy" (/woman). She clearly needs it to be the second - and doesn't like to lose. Tough on her.
Obviously, you should take care that it doesn't just become "when I'm over at dad's place, I can do whatever I want, it's happy fun time, no rules! Whoo!". When my nephews or nieces stay over here, it's easy for us to allow them more freedom etc than they have at home - they're only here for one or two weekends a year, it's a treat, we're going to the zoo and museums and whatever. As a co-parent, that's a trap to avoid...But from what I've read and heard so far, I think you're doing a pretty good job of avoiding it.