I think the quip is intended to say that Berlin was adept at writing music in the "Black" idiom—or at least as that idiom was perceived at the time. People thought his music had the ring of authenticity. However, this sense of authenticity could be *separated* from the individual (Berlin). He was not perceived to *embody* Blackness. It was one of the "tricks up his sleeve." (in his closet ~ up his sleeve).

There is a racist logic to why the person in the closet was a boy rather than a grown adult. I don't know whether you want to explore that 'logic', but I believe at least, intuitively, that I understand why people did that. That is, why at the time racists —who weren't all about Racism, but who were racist nonetheless— used images of Black kids.

What I am trying to say is that I think the fact that the person in the closet was a boy is merely incidental (i.e. incidental to the language of racism people were using). The main point was to note Berlin's skills of (as it was perceived) imitation. And to say it in such a way as made clear that "Black" stuff was indeed both distinct in its characteristics AND *district from* the (perceived) essential make-up of Jews and/or Whites. It's a comment that rejects the idea that anybody can learn anything and that culture is independent of race. In later eras, it would be looked at as a good thing if a White/Jewish performer had "grown up with" African-American culture because that would provide an elegant way to account for the authenticity of their music. This quote, however, reveals that while people appreciated the "Black" characteristic in the music, they did not want to view the performer as partly Black himself.