Why can a (industrial) servo motor run faster with higher DC-link voltage
But for the speed, it depends which frequency the inverter delivers to
  the phases of the machine.


That's just partially true. The voltage is the one which determines the speed, like for any brush DC motor. The frequency has to vary (or voltage VS. frequency, doesn't matter) according to setpoint voltage with V/f characteristics.

Basically the voltage rating of the driver is related to the mains supply voltage, standard values are 120V, 200V, 400V, 480V. Then within the same voltage rating, they differ in current ratings. You can combine the servo motor and appropriate driver as manufacturer declares. Not always the entire voltage range is used, some motors require high current, but low voltage, those are high dynamics motors. Some other might need higher  voltage and lower current, those are high inertia motors.

The simplified answer: Think as you would have brush DC motor, as it is easier to understand. Current means torque, voltage means speed. The frequency is just a consequence of AC voltage instead of DC.