What is this feature on a Piaggio P.180?
I believe it's called a dolphin tail, but why is it exactly there and what is it for and why don't all airplanes have this?
The airfoils shown are delta fins. They provide lateral stability during higher speed flight. It helps to reduce dutch rolling tendencies and to prevent deep stall characteristics.
So much so that on the Learjets I fly, there is no need for an operable yaw damper. There is also no stick pusher or stick puller system. Older Learjets without delta fins are required to have those two systems installed.