Rotational dynamics about a point or an axis?

The discussion is mostly semantic. They are both calculated relative to a point, in the case of the torque the point has the additional meaning that if you put an axle trough the point, the object will start to rotatte around it if the net torque is not zero. It happens also that the torque will be the same if you chose any other point along the axis (axel).

Well for a rotation the reference taken is the axis not a point as for any system of particles, given a point, it can be rotated in many different ways so to define a unique rotation you choose an axis, and that applies to both angular momentum and torque. But let's say you want to calculate the angular momentum of the earth as it is rotating and simultaneously revolving, so both motions would contribute to angular momentum and for the revolution part the angular momentum would be defined about a point but the one associated with rotation would be defined about the axis.