Torque, for a system of particles, is defined as:
$$\boldsymbol {\tau}_{net} = \sum_{i=1}^n \mathbf r_i \times \mathbf F_{i,net}$$
Here $\mathbf r_i$ is the position vector of the point from the coordinates axis. Now usually this coordinate axis coincides with the axis of rotation.
Torques exist independent of rotations. Indeed, in statics problems nothing is rotating, yet we can still choose points of reference to calculate torques about (and show that the net torque about any such point is $0$).
You just pick some reference point, then you apply your definition to determine the torque caused by forces about that point. Of course for certain analysis it is smart to choose the point that coincides with the axis of rotation, but this is not a requirement to apply the definition of torque itself.
This is also found in the definition of angular momentum $\mathbf L=\mathbf r\times\mathbf p$. You can choose any reference point you want to calculate angular momentum about. And you can still write out $\boldsymbol\tau=\text d\mathbf L/\text dt$ about that axis, where the torque and angular momentum are relative to the same axis.
A fixed rotation axis, such as with a car wheel, can only rotate around the axle. An unrestrained object will only rotate about a line through it's center of mass. Applying a force off-center of it's COM will cause rotation and translation.
