This reminds me of a physics professor in university. He was Indian, and when I saw him on the first day of the course, I thought I would have difficult time with already difficult course material, due to the incomprehendable accent. To my shock, the professor introduced himself with perfect Oxford accent, and even wrote pronunciation of his name next to his written name on the chalk board. He was as British as Her Majesty the Queen.
I agree with that idea. Language is central to identity.
my father isn't a native English speaker but you would be hard pressed to find a slip or pick a specific accent out of it.