Continuity is part of it, but not all of it. Ditto for appearance on film: the prototypical Chinese food box is a piece of visual shorthand that instantly communicates what’s going on. If they had another type of takeout container, like those big styrofoam cups with the translucent plastic lid, the audience wouldn’t know immediately what they represented, and there’d be a moment of distraction while they worked out what they were looking at, during which time some dialogue or other information might be missed.
There’s another reason: The boxes are closed, so you don’t actually need to have food in them. For all the difference it makes, you can fill most of them with sand so they don’t move around, and have food in just one or two, whichever ones the characters are snagging their gyoza out of.
This is important for a couple of reasons. First, expense. Sure, a big Hollywood movie spends more in three months getting bottled water for its diva star than most of us spend on a brand-new car, so a few bucks for a spread of food isn’t going to break anybody. Some movies watch their pennies, though, which makes these takeout boxes a nice shortcut.
Beyond that, though, a two-page dialogue scene with four or five people around a table can take a whole day to shoot, or even longer if it’s a big Hollywood movie. You’ve got to get a master shot, or two masters if you’re getting coverage from a couple of angles. You’ve got to have closeups for every single important character at the table. You’ll have middle-ground shots, which include two people rather than a single closeup. And the director may want something a little fancy, like that shot in The Untouchables where the camera makes a slow circle completely around the table. Every single one of these shots must be lit and rehearsed, which takes a lot of time. Then you have multiple takes, which can be from three to fifty depending on the director, the complexity of the shot, and so on. In the end, this one- or two-minute scene can take eight to sixteen hours to put on film.
And in that time, the food is just going to be sitting there. If you’ve got strong lights (which is less common now, with faster film and digital alternatives), the food can start to go bad pretty quickly. (Cold foods melt fast and are substituted; mashed potatoes are used instead of ice cream, for example.) That’s especially true for something that takes more than one day to shoot. Even if it’s just six hours on the set, it’s still gonna start to smell.
So you minimize that by having just a couple of boxes of food, and you put a bunch more boxes with rocks or whatever in them, and voila, instant fake feast, in which you don’t have to spend a lot of money, you don’t worry about continuity, you don’t stink up the set, and you don’t ask the actors to eat food that’s been sitting and spoiling for a few hours.