For example, Grab brush isn't just called "Grab." It's called "Grab - Shot Sculpt." Same with "Scale – Shot Sculpt." Does it mean these brushes are fundamentally different from the others?
Poisson Ratio When a Shot Sculpt brush is on, controls how much the material should move in the direction perpendicular to the load (like how much a rod narrows as it is stretched). See the Wikipedia page for more info.
What does this mean? "material"? But doesn't Sculpt SOP just an SOP, which works on the geometry? How does it affect material?
I believe it's 'material' as in the 'substance' or 'matter' you are deforming, so it has nothing to do with shader-materials. The wording can indeed be a bit confusing.
Instead, Poisson's ratio will give the brushes a different (often nice, organic) feel. The wiki provides the mathematical explanation, but simply put, it creates a thinning effect when stretched and a bulging effect when compressed:
I'm not entirely sure why the brushes are called 'shot-sculpt,' though. As far as I know, it does not involve sculpting over multiple timesteps.