It is a frequent idiom for a group of characters to interact while eating Chinese food. Generally of the carry out variety, with chop sticks being plunged into the deep fold-up boxes. It seems to me that this is intended to bring a particular thematic element, but I can't quite put my finger on it. I see it way too often for it not to have a specific significance.
I do think you have something in thinking that Chinese food has a particular significance as take-out. There's an aura of a comfortable, but pressured, environment, where people are banding together to get stuff done. But, enough of my personal opinion and on to some quotes and particular examples. Much of this is cribbed from TV Tropes.
- Jewish people and Chinese food: Starting with the trope of Jewish people going out for Chinese on Christmas (Peking Duck Christmas), there is a broader association in all cases. For example, in My Favorite Year, the more urbane of two put-upon assistants bonds with the romantic interest by ordering Chinese food when they have to work through the night, saying "Catherine, Jews know two things: suffering, and where to find great Chinese food."
- Chinese food and long hours/stuff to do: This is particularly associated with The West Wing where the characters order Chinese food when they're facing a long creative challenge, like writing jokes for the Correspondent's Dinner
- Chinese food and comfort: Orange is the New Black uses the theme this way by centering the main character's romantic choices around a quote about Chinese food. After having a wild and tempestuous relationship with Alex, a drug smuggler, the main character Piper is told by her more settled best friend that eventually, she's going to want "someone who knows when to order Chinese food".
In summary, Chinese food is associated with familiar but stressed situations, particularly among more intellectual and/or Jewish characters; and can imply both "something needs to be done" and "something needs to get fixed emotionally".
I think sometimes people look for symbolism a little too much in cinema. In this case, I contend that there is no symbolism in Chinese food. It just happens to be an easily recognizable take-out food that can be recognized without branding. Also, hamburgers or pizza have been used but there are a lot of challenges with that. You have to control the environment, and when someone takes a bite you can't just replace it. But if you throw some rice or noodles in a box and someone takes a bite, you can refill it.
I don't think Chinese food, in particular, is being used as a theme. I think it's delivery food in general. It evokes the sense that these people don't have time to go anywhere or do anything other than work on the project they're currently working on.
In America, there are really only two universal foods that are delivered: pizza and Chinese. That is changing, and it varies by region, but those are the two standards.
So, the question then would be, "Why choose Chinese over pizza?"
- Sometimes they don't. Sometimes it's pizza.
- Some people might considered Chinese a "higher class" of food than a greasy pizza.
- (My personal opinion) When the characters have that A-HA! moment, it's much more effective visually to point a pair of chopsticks while saying, "Yes! That's exactly it!" than it is to point a floppy piece of pizza.
I think that movies and TV shows use those Chinese takeout boxes just because it’s easy for production. They could be empty boxes and they wouldn't actually have to prepare any food for the scene or multiple scenes at that.