What do you think lahiri means by this?

the traditions on either side of the hyphen dwell in me like siblings

The goal of the protagonist is to learn every single secret in chess.

Below are some of the excerpts from the story:

"I went to school, then directly home to learn new chess secrets, cleverly concealed advantages, more escape routes" ("Rules Of The Game", Amy Tan, page 7).

"I borrowed books from the Chinatown library. I studied each chess piece, trying to absorb the power each contained" ("Rules Of The Game", Amy Tan, page 4).

The story explains and shows the motivation the protagonist, Waverly Place Jong has.

Explanation:

"The Rules Of The Game" is a story by an American writer, Amy Ruth Tan. The story opens with a focus on silence and teaches how having control over one’s emotions truly endows one with a secret strength. Waverly and her mother, Lindo Jong, tend to have a psychological battle, each of them was trying to gain the upper hand over the other. But one important rule is that one must remain silent in order to emerge as a winner.