Non-Breath Oblige is about many things- cancel culture, human nature's drive to pass blame over accepting responsibility, scapegoats, and bullying...
Those who hold their breath have become the targets that allow society to continue.
Holding one's breath in the song means to keep quiet, and accept your position as a scapegoat. Fighting against this system would only rock the boat and bring even more shame.
The phrase "Non-Breath Oblige" is a play on the French phrase "Noblesse Oblge", which refers to the responsibility of privileged people to act with generosity and nobility toward those less privileged. In this song, it is used ironically, as the responsibility of the oppressed is to keep quiet and bear their appointed fate.