[semi-extrinsic]

No, the handbrake (not emergency brake) is actually connected (mechanically) to a completely separate set of brakes on the rear wheels. Only on custom-built drift cars á la Ken Block is the handbrake connected to the normal brake calipers.

[qbrass]

It depends on the car. Some of them engage the same calipers the hydraulic system uses. Some have a separate caliper or drum. Older, drum brake cars engaged the same shoes the hydraulic system used. And if you go back far enough, some had mechanical pawls or band type brakes that engaged at the transmission.

And drift cars use a separate hydraulic brake attached to the rear disks.

[refurb]

This is correct. Some emergency brakes uses a screw mechanism to activate the brake cylinder piston to push on the pads. Same stopping mechanism as if you push on the brake pedal.

Others (usually less expensive cars) have a set of drum brakes inside the disk brake that act as emergency brakes.

If you have rear drum brakes, it's the same as my first example. The emergency brake activates the normal braking system.