In a plane in level controlled flight, there is very little to hand off between two type-rated pilots. Both folks understand the machine, and the machine is working the way it is supposed to. Humans work the same way! Parents "hand off" their healthy kids to schools or babysitters or relatives every day.
But imagine a plane that is in the process of crashing; it's in a dive, controls are not responding as expected, one of the engines keeps turning off. A pilot is fighting to regain control... how comfy are you with THAT pilot handing off the aircraft to another pilot in the middle of that situation?
It's a little silly as an analogy, since plane crashes tend to be resolved pretty quickly one or the other. But conceptually, just imagine a plane that is in the process of maybe crashing for 12 hours. There's a good argument for a pilot to just see that through instead of "clocking out" at 8 hours.
Are you aware that on long haul international flights pilots do in fact rotate who is actively "flying" the plane? Flying is in quotes because most of the work is done by automation these days. I won't draw any analogies between the autopilot doing much of the work for pilots and nurses doing it for doctors because I can't actually support the statement with any data.
I should also point out that the vast majority of doctors don't work 120 hours a week continuously, rather, they experience higher-than-usual clusters of working time vs. not working time. That is, they might be on rotation for 36 hours straight but then off for 36 hours or more. And some of their shifts might only be 8 hours. Residents work longer hours, but are supervised by doctors.
I know several ER docs and they all cite the dangers of patient hand-off as the main reason they continue to support long shifts.