I have watched several Chinese and other Asians try to keep there children speaking their own language. But often they loose the battle. The parents speak Chinese at home but as the children get older they start to only hear Chinese but speak back to their parents in English. If the parents work in an English office they become fluent as well, but often with a heavy accent. The kids lose their accent or not depending on their school friends (and number of siblings). If their school friends all have similar backgrounds they can keep their native tongue longer. But if not they lose their accent and soon lose their ability to speak the language well, even if they can understand it.
I’m guessing your wife was in a Chinese community here. With Chinese friends at school and maybe Chinese Sunday School. The large family helps keep the language and if the mother works in an environment that only speaks Chinese she will be less tolerant of English spoken in the home.
Anyway the Chinese will separate your wife as not being one of them. Go to Shanghai or Xian. Banana, twinkie, ABC are all the same meaning. As much as we will still hear the accent and some of the old ways. Those back in China will know she is different. Besides her accent, she will have more confidence but she will be less competitive. She will be more likely to small talk and laugh. And the next generation really will be every bit American as anyone from Europe or anywhere else.
The experience on an American school playground and the entertainment media here often put a defining mark on a child as American.
Oh, we're well aware of that! Sometimes I think my wife acts racist, the way she acts towards FOBs (Fresh-Off-the-Boat or Chinese from Asia). She hates foreign Chinese not born here in the USA. And it goes both ways, the FOB Chinese show disrespect towards Chinese-Americans born here. Different culture, she thinks the FOBs are rude. Anyway, she was born in San Francisco's Chinatown, and her parents moved to a white neighborhood in SF when she was young, maybe 4. They only spoke Chinese at home, even though her dad had been here since he was a kid and was even in the US Army during WWII fighting in Europe for the USA, spoke English outside the home.
My wife was an IT manager of a large data center, and had some conflicts with Chinese employees. Because of culture differences, they couldn't stand working for a woman. One Chinese guy she fired did a lawsuit against her and the company claiming racial discrimination. The lawsuit went forward a while because my wife uses her married name (mine), then when lawyers found out she is Chinese the guy's lawsuit fell apart! Guy had to be dragged out of the offices by SFPD when my wife fired him. She became known as the Dragon Lady Boss because of that. She had white, black, hispanic and Asian employees under her, treating them all the same, although she disliked the Chinese guys and their anti-woman boss bias.
I've seen similar stuff in my job sites. One of the toughest managers was a Chinese-American gal from Texas, she took no crap from Chinese male employees and had them cowering in fear of her. I liked seeing her do that! On a different note, I've had many bosses over the years, and some of the most incompetent ones were either Chinese men or Black (both genders). Maybe I was just unfortunate, but I had better success with white bosses, again I'm generalizing about my experience.