Only a tiger fights like a tiger.
When a druid has Shapeshifted into a tiger and uses Hack and Slash, they aren't fighting like a tiger. They're fighting like a druid who's turned into a tiger. So they deal druid damage, which is 1d6. Various advanced moves like Red of Tooth and Claw and Formshaper improve their baseline combat capabilities, perhaps by bringing their basic mindset closer to that of the animal.
But similarly, it's not burning any hold to Hack and Slash as a tiger, because they're not tapping into any of the tiger instincts. Only animal moves that let the instincts take over actually burn off the hold from Shapeshift. Now, the results of animal moves can look a lot like the results of player moves - one of my fallbacks to handle an animal that attacks ferociously is to deal +1d6 damage and get opened up to an attack - but for all that a player move might align with an animal move, if they don't burn hold, they don't tap into the animal instinct and get no special benefit.
Now, they are the size and shape of a tiger, which means that they may be capable of fighting in melee with things the druid themselves cannot. And the forms a druid can turn into with, say, Thing-Talker or World-Talker may be dangerous to touch in and of themselves. But these are considerations of fictional positioning and aftereffects, rather than raw damage.