As a Dane from central Copenhagen, I'm not sure where you'd get that idea from.
While most would agree that fresh food tastes better (unless it's pizza, that jury is out on that one), that doesn't mean we're too posh for the looks of leftovers. A good portion of our food is also quite boring, and the most common takeaway comes pretty crudely presented and packaged. If we only ate food in fancy presentation, we'd all have starved to death by now.
The reason I don't eat half-eaten meals is because I never half-eat meals in the first place. I order the amount of food I want to/can eat, and should that plan fail, I share with my tablemate(s) - although I tend to be the recipient on that one. We don't do it because we fear taking a crime-scene with us home in the form of a haphazardly filled styrofoam box like poor people, but just because it seems polite and proper to both size your mail right and try finish it. That, and that we didn't know that taking it with us home was even an option in the first place.