Are they all just called anbu or do they have special divisions like the hidden leaf anbu?
ANBU means Ansatsu Senjutsu Tokushu Butai(Special Assassination and Tactical Squad). It is a common name for any village.
As described in ANBU wikia,
ANBU are covert operatives dispatched by their Kage.
Anbu are recruited from their village's standard shinobi forces, hand-picked by their Kage for their individual capabilities and special skills.
As described in ANBU's uniform, the outfit are different but not what they are called. It is more like Kage which like post or title.
And yes they do have special division like Hunter-nin from The Village Hidden in the Mist.