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First, and I promise I'm not trying to be pedantic, it's worth noting that not all VFDs can run a servo motor. They're more typically referred to as servo drives.

Traditionally, synchronous AC servo motors rely on some sort of feedback device that indicates the position of the rotor. It uses the rotor position in conjunction with the drive's internal motor model to vary the frequency, voltage and current to commutate the motor and match the setpoint.

In an AC servo, torque is directly proportional to current. Several manufacturers list this constant as Kt (torque constant). It is often expressed in newton-meters per ampere (Nm/A). Motor speed is directly proportional to voltage and this is often referred to to as Ke (voltage constant). This is often expressed in volts per thousand RPM (V/kRPM).

If there's a mismatch with these parameters, you will typically see very poor performance of the motor. It likely will be unable to develop the required torque or speed. If one really screws up the parameterization, you can burn out the windings with too much voltage or demagnetize the motor by dumping too much current into it.

When I was tuning motors in my previous life, I would NEVER exceed the listed peak (instantaneous) current or torque. We did very often exceed the continuous rating as our duty cycles were quite low but this is fine as long as you're in the proper area of the torque curve for the duty cycle.

What determines the current draw is a variety of things. Obviously, the load that the motor is trying to move is a big determining factor. But manufacturers build different size motors in the same frame size to trade speed for torque. They do this by changing the motor windings which, in turn, changes the stator length (how long the motor is).

There is a newer method known as rotor flux control that uses very sophisticated motor modeling and analysis of the back voltage and current to commutate the motor without an encoder but I have no experience with it.

If you REALLY want a great breakdown of how all of this stuff really works, find a copy of the Control Techniques Drives and Controls Handbook. It's expensive, but it is a fantastic book. Amazon link.