Is the mechanical center not necessarily the same as the electrical center?

Correct.

Can they be vastly different?

I've heard people say they can center their servos mechanically but I'm not finding that to be the case.

Electrical would be related to the signal Tx sends to center servo.

Mechanical center is what you try to physically adjust using servo arms/wheels in relation to your setup needs and if you cant exactly find it, thats where "servo subtrim" in Tx comes into play.

~~Enjoying the hobby one flight at a time~~

I'm probably not using the correct terminology.

By "mechanical" I meant if you spin the servo by hand and find the center of the stroke (without power of course).

This will not get you the same position as you get when you power on the servo. Correct?

Got it.

In theory it should provided its perfectly engineered and all electronics are performing flawlessly.

There are "servos testers" to make up for the real world side of things. Specially when ganging them up to perform a task in unison.

EDITED:
Tx calibration also plays a role in centering. This is usually noticed when once servo just wont center in one Rx port but does on another.

~~Enjoying the hobby one flight at a time~~

"By "mechanical" I meant if you spin the servo by hand and find the center of the stroke (without power of course).

This will not get you the same position as you get when you power on the servo. Correct?"

Not necessarily.

There is a mechanical limit built into the server - this is the hard-stop that you find by turning the servo wheel while the servo is unpowered. The center point between the two mechanical limits is the mechanical center of the servo travel.

Inside the servo, attached to the servo output wheel, there is also a potentiometer (pot) that electrically feeds back to the RX where the servo wheel is positioned. The center of the pot sets the electrical center of the servo travel.

If the pot is not aligned properly, the electrical center is not going to match the mechanical center. It is the electrical center that is important to setting up your servo / linkage.

Additionally, a dirty pot will cause the servo to jitter and slip at the center - I have replaced 3 Align DS410M servos because of dirty pots.