I feel like there is something I am supposed to know that would help the author make more sense about the flute playing. But I don't really understand what he is talking about?

Was this episode in childhood and now he is a young adult? Did he attend public school?

I guess flutes and to a lesser extent clarinets were seen as "girly" (or maybe gay) instruments and boys were bullied for playing them.

The association isn't new, see the related slur for gay people.

I think the main point is that

1. it's unfair that boys who chose to play flute got bullied for it

2. it's not good that his brother essentially took his choice from playing flute by reinforcing #1

3. #2 is actually simultaneously the way to minimize the negative effects of the authors choice of instruments

So in spite of those first two things individually being pretty shitty, they actually give you an optimal outcome (socially) in school. Ideally we get to a place where #1 stops happening, but you won't solve it by going against the grain, you'll just be a martyr for nothing.

This was about a middle school kid choosing their instrument for band class, in their case they played through high school.