How is the feedback estimated in the torque mode control of a servo motor?
Is the product KtI of the torque constant Kt with the measured
(perhaps filtered to denoise) motor current I the only method to
estimate the current generated torque in real-time?

You are asking whether the current measurement is the only way to estimate current. The answer is that it is the most straightforward method.

Additionally, is the position-velocity-current cascade control loop
schematic shown on the linked webpage in error, since it does not
explicitly illustrate the torque feedback?

This one?

It's ok, since the box Drive gets a current command and it outputs current, so this is a torque/current controller. It's usually a PI controller with current feedback.

Finally, if the schematic is indeed accurate, then is it correct to
say that a motor can be run in position mode or torque mode at a time
and not both at the same time?

Basically you are correct. But it can also do positioning with reduced torque, if the current command is limited to max/min value.
But usually an industrial  servo drive is a cascade of at least velocity and current controller, so the input is a velocity command and the encoder feedback is connected to it. It can switch the topology of the cascade, so that if the torque mode is enabled, the velocity controller is disabled but the torque is limited (in both situations: velocity/current or extended: position) for maximum speed \$\omega_{max}\$, maximum inverter power \$P=V\cdot I, \;T_{max}=k_t\dfrac{P_{max}}{V}\$, maximum motor current \$T_{max}=k_t\cdot I_{max}\$, maximum motor power \$T_{max}=\dfrac{P_{max}}{\omega}\$.
EDIT:
The pre-process of torque/current setpoint, the input is the speed controller output  (M=Torque), output goes to current controller input.

Speed controller (PI) with feed-forward path for friction, inertia, dead weight.