Does servo.write() function use different channels for different servos?

On some servo shields, you can split the current draw from servos through different channels so that two or more servos don't draw current at the time, which allows you to use less amperage for powering your servos. These usually come with custom libraries where you have to designate the channels for which servo to use.

I am wondering if there is a similar thing in the default arduino library, or if servo.write() automatically switches between different "channels" when multiple servos are drawing current so as to reduce maximum amperage required to power the set of servos.

Servo.write only sends control pulses - it knows nothing about power supply arrangements.

Whether a servo is drawing current depends on the mechanical load, not the commanded position.