Is Torque always defined about the axis of rotation?

Question

My professor told me torque can be calculated about any point in space, but would such a torque ever make sense even if the particle is rotating about some other axis, on which the point does not lie?

Answer 1

but would such a torque ever make sense even if the particle is rotating about some other axis, on which the point does not lie?

Yes. Both calculations should give the same result (assuming a non-accelerating object). But it may be much harder to calculate.

Forces on the line through the point of reference create no torque. So if there is an axle or a hinge there, we can ignore it for torque about that point.

If we consider a different point instead, the forces at the axle or hinge are now relevant (and possibly unknown).