First of all, let me tell you how much I appreciated the way you are reaching out to the student in order to deal with this issue. Schooling is usually the first taste of the world out there, for children, and I think that assigning you to tutor him was a very sensible decision too.
I agree that his discomfort could be caused by any of the reasons that you mention, whether in his surroundings or within himself. You might be able to help him overcome his unease by pointing out that you were once in the same boat. He is likely to look up to you, as his teacher, and a key to the language secrets that he wants to unlock. He needs to understand that people are not what they are out of the blue, there is work to be done and progress to be achieved for us to become what we want to be.
I don’t know exactly how old he is, but he may be able to understand that “speaking with an accent” is what we all do. There is no acent-less speech, so you can’t “get rid of an accent” without acquiring a different one. You may use teachers and children at the school, or radio/TV programmes, to exemplify different accents and different ways of speaking. He seems to be quite aware of how people sound, so he is likely to respond to examples like these. Everyone looks different too.
Another approach could be making it clear to him that the accent that he has right now is a natural part of what he is right now. It need not be a permanent part of him for that. He can make changes to it (like you presumably did too?), but only if he speaks the language as much as possible, so that he hears himself and can match what he sounds like with what he wants to sound like. Try getting him to tell you who exactly has the accent that he wants to have? He must have some role model in mind. You and him can then work (or play) towards that accent little by little. His keen ear will be invaluable help to him here, and I predict quick results. That will hopefully boost his confidence in his abilities.
One last thing: I also agree that his family are not the key to his development in English. He needs their support for his home language, because being multilingual is also a natural part of what he is.
Do feel free to contact me privately, if you wish to discuss these matters in greater detail.
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