The axis of rotation of the earth passes through the geographical poles or is there a true axis?
Geographic axis is NOT fixed.

North and south poles are NOT fixed.

Ideology is reverse of what you are thinking. 

First suppose that Earth is rotating about some arbitrary axis => Then this axis intersects on Earth's surface at two points => Now, if you see earth from above and observe earth rotating in counter-clockwise direction then this point is defined as North pole and hence other will be south pole.

In short, first we see earth rotating about an axis and then north and south poles are defined according to it.

Also, the axis of rotation wobbles and as a result north and south pole wanders a bit. This is known as Chandler wobble.

Note: It is very important to realize that according to chandler wobble axis of rotation wobbles with respect to Earth. Earth itself is NOT wobbling due to chandler wobble, it is only changing the axis of rotation.

So, there is no concept of true axis here, but only one axis -> geographic axis which wobbles.