After years of juggling between 2 accents for 2 languages. All I'll say is that its weird. Unfortunately, I'm not bold enough to "not care" about it so I will probably continue to do so for the foreseeable future.
The Chinese accents on Everything Everywhere All At Once are what I identify with most. Michelle Yeoh’s Malaysian Cantonese inflections in her Mandarin, Ky’s Vietnamese Chinese accents in his Mandarin and Cantonese, makes me connect with it more than something like The Farewell
On one hand, speaking Malaysian accent in US leads to, "oh where are you from, oh yea I can hear your accent", or you might be made fun of your accent, or just "yea your accent is wrong". 😶
On the other hand, speaking US English accent or Northern Chinese accent in Malaysia is seen as pretentious most of the time because "you're too good for Malaysian accent". 🙃
The little silver lining is when you randomly recognize other Malaysians / Singaporeans on the street just from their accents (especially when lining up at some South East Asian restaurant).
Anyway, I believe I'm not alone in this situation / dilemma. Some who moved to a different country learnt to adapt to the new accent (and some even left their old accent behind). I know someone who stopped talking to old friends because they would make fun of their US accent.
I remember hearing someone else once said this about accents that I thought its worth sharing, everyone coming back from UK, US, Australia learnt a new accent but no one ever came back from India with the Indian English accent.
Oh, almost forgot about this. Here's a somewhat relevant funny video from over 8 years ago by @dmingthing.