I'm not sure if the stomach actually does drop a bit when you're taking off in an airplane or when the airplane falls but I can say that when I fly on airplanes and the airplane is taking off it does feel like my stomach is dropping until the plane gets into the air.
I've also experienced turbulence and during turbulence my stomach felt like it was dropping or had dropped but it quickly went away after the turbulence went away.
The sinking feeling in my stomach after taking off in the airplane went away within 10 minutes of getting into the air.
Not everyone feels a sinking feeling from what I've heard but it's normal for some people to get a sinking feeling or feel that their stomach has dropped on the plane.