Does Pawni deal extra damage to broken enemies?

The condition has two doors, not one. Break is the obvious trigger, but any crowd control effect opens the same window, so the bonus is live more often than a break-only trait would be. Pursuit is a passive. You do not press anything to turn it on, which means the whole skill is really a scheduling problem: land the control first, then commit Pawni's damage into it.

That points straight at team building. The standard early composition is a DPS unit, a break unit, and a healer or support, and Pawni is the piece that makes the break worth doing rather than a stall. Support Aniimo apply buffs that amplify outgoing damage, so a support slot pulls double duty here, feeding the same window the break opens. Role labels are loose in this game anyway. What a creature actually does comes from its skills, so build around the trigger rather than the label.

When the bonus works against you

Breaking a wild Aniimo also improves your chance to catch it, which puts Pawni in an awkward spot during captures. You want the target broken and you want it weakened, but you do not want it dead, and a bonus that fires exactly when the enemy is broken is the fastest way to overshoot. Catch stance covers this. An active companion in catch mode stops attacking once the wild target's health drops below a threshold, so switching Pawni into it before the break lands is the difference between a capture and a corpse.

Against Omega Aniimo none of that applies. They cannot be captured, so there is no reason to hold back and the trait is pure upside.

Not settled: The trait is written as a ceiling, up to 30%, rather than a flat bonus, and what moves it between nothing and the full value is not settled yet.