Strange find - but Dealer/Toyota 800 could not answer. Found a piece of Carpet sewn in to the Leather Drivers Seat on Prius 4. Location is on the right side down by the seat between console. It's like 12" Long right below place where you put seatbelt in and extends to the floor carpet down about 6". It looks like it blocks out change or pens, whatever from falling under the seat - BUT - noticed there is no such thing on the passenger front seat leather cushion left side next to console. Only thing I can guess now is - there are a lot more electronics under the drivers seat and closer to console side - so I think reaching down fishing for change or whatever might disconnect something. On the passenger side only a single box with 1 main wire but towards the door side. SO what is it and what is it for?
Prevents leather wear (that part of the "leather" is actually vinyl) and the squeaks it would cause as the seat moves slightly against the console. Quite common in vehicles.
The piece of carpet like material between the driver's seat & console is a deflector for some objects that might cause a problem with the electronic chasis under the seat. The driver's side seat has a wide clearance on the console side, I don't believe it is for stopping a squeak!
It may have to do with the exact characteristics of the driver's seat which may have casued a problem resulting in the fabric modification of the drivers seat alone.
I've seen pieces of carpet used there and under the seat bottom to absorb wear, and even inside the seat back. It just wears better and is quieter than any other solution that costs less than a fortune. Back when I used to adapt seats from wreckers to 4X4s. Those days are over. With airbags and other restraints, it's dangerous and potentially expensive to mess with seats.