[carguychris]

That "thing" is categorically known as a grip adapter. The one on this particular gun appears to be a Tyler T-Grip. These are made of cast aluminum and have a strip of thin copper riveted to the inside. To install the adapter, the shooter removes the stocks (aka grips), wraps the copper strip tightly around the frontstrap, then reinstalls the stocks. Friction holds the copper strip, and therefore the grip adapter, in place.

IIRC Tyler also used to make a version without the finger grooves, but it has been out of production for decades. Pachmayr used to offer plastic grip adapters with no finger grooves; these usually had vertical serrations along the grip face and were offered in 3 sizes. They have also been discontinued for a long time.

S&W once offered a 3-piece factory adapter that clamped around both sides of the entire grip and had a hole for the screw to pass through; however, these adapters are quite rare.