I have become a fan of music since then is wrong or it very specifically means that something happened between the time and now (I guess that is not what you mean). If we are using 'since' we would not use it with become. I would say either of these: I became a fan of music then I have been a fan of music since then

They're not quite the same. Let's say that 'then' refers to a concert that you went to last year. "I've been a fan of music since then" means that your transformation from non-fan to fan occurred at that moment in the past. When you were at that event you suddenly became a music fan. Your newly-found enthusiasm for music began at that concert. "I've become a fan of music since then" means that the transformation happened during the period between the concert and now. Perhaps the concert made you start thinking about music, and then afterwards you began listening to more music and gradually became a fan at a later point.

No, they are not the same. have become - means it is happening now and will continue in the future. have been - means it happened in the passed and is still continuing. I hope this does not confuse you.