How Dual MOSFETs are used in Power Management or Protection circuits?
Here comes the problem. why the left MOSFET turns on when its
drain-source voltage is &quot;-1V&quot;?

It's not a problem; MOSFETs are 4 quadrant devices in that they can fully turn on even when the drain source voltage is reversed providing the gate voltage is sufficiently more positive (NMOS) than the source voltage: -

Image from here.

the MOSFET only turns on when the drain source voltage is above a
limit

Not true.