Range of angle in axis-angle representation of rotations

According to Euler, one can represent any rotation in 3D by an angle in the range $[0,\pi]$ and a unit vector representing the direction of an axis of rotation, some details are here. Other possible types of representation of rotations include the rotation matrix formalism, Euler angles, quaternions, etc.

I am struggling with understanding of why the range of an angle in the axis-angle representation is $[0,\pi]$.

Let's say in this example we turn by $3\pi/2$, what is the corresponding angle in the range $[0,\pi]$ along the Euler axis $[0,0,1]$? Or should we choose another axis? Thanks!

A rotation through an angle of $\frac {3\pi}{2}$ about the axis $[0, 0, 1]$ is the same as a rotation through an angle of $\frac {\pi}{2}$ about the axis $[0, 0, -1]$.