The question of "what genre" is a bit odd, as the answer is whatever speaks to you, as the designer. There are genres that might be less appropriate, (like FPSs) and genres that might be more so (like simulations), but the field is still wide open.
However, there are still considerations when wanting to drive an experience through game design, especially one so untouched as cultural assimilation. What's important here is to dissect the process and finding which parts of cultural assimilation are trade-offs, compromises, or other kinds of "interesting decisions" an immigrant must make.
The most obvious of these, of course, is the straightforward "The more you assimilate, the less you adhere, identify with, and even remember the home culture." Assimilation allows for support systems, relationships, a sense of familiarity, comfort, and security. Traditions and old cultural ties are a source of identity. Any game mechanics should, mechanically, contrast the need for relationships, jobs, social support, local services, and a sense of belonging with the sense of personal identity and need to stay connected to the old.
I can't, obviously, give any specific details, because there is no definitive mechanic (for this or any other concept). This is where the designer comes in and determines how to best serve the contrast of social and financial security with personal identity. But the important note is that the contrast and the balance of those two are where you'll find the mechanics telling the experience of cultural assimilation.