IME, no second-generation Asian cares enough about ethnic backgrounds to the point of being awkward like this. First-generation Asians sometimes do have beef, but they also don't act like this either. They tend to guess the ethnic background based on last name, the particular accent of their native tongue and English, what church or temple they go to, what city/part of the city they live in, and so on. They don't have to directly ask the ethnic background of the other person, and if they care to reenact the beef in a diasporic context, they mostly just act standoffish and talk shit behind their backs instead of being awkward like this.

So, in this hypothetical example, the first person would speak either in Putonghua (standard Mandarin spoken in the Mainland) with erhua or Mandarin with a Cantonese accent. The second person would speak Guoyu (standard Mandarin spoken in Taiwan) with a Hokkien accent since benshengren are more likely to identify as Taiwanese than Chinese. Their names would be pretty different and give away their background as well. The first person would probably have a name like Chen Li while the second person would have something like Alice Chang owning to the Taiwanese diaspora's tendency to more readily adapt English first names. Chen Li would go to the mainlander church, and Alice Chang would go to the benshengren church, so knowing what church they go to is enough to reveal their background without them saying a single word.