Does taming a brat mean punishing them?

Only in the sense that both people find funny or hot. The point of brat play is the contest, not the consequence, and a tamer who treats it as genuine discipline has usually misread the game.

What good tamers actually do is pay very close attention behind a relaxed exterior, so they can tell the difference between resistance that means keep going and resistance that means stop.

If the brat stops enjoying being caught, there's no game left to win. That's the whole test of whether it's working.