What do you call the part that the shift gear cable is attach to along with the stick shift under the arm console?

That section is called the gear shift linkage. Most likely, the clutch cable ends in an "end loop", which is a circle on the end of the clutch cable. There's a metal ball at the bottom of the shift lever (the gear shift, the thing you change gears with) that goes into the end loop. This metal ball going into the end loop is called "Ball Joint Linkage". The little fastener that keeps the ball and the end joint together is called a bushing.

Don't let a mechanic replace your clutch cable when all you have is a broken bushing!