How can earthquakes shift the earth's axis?

Your explanation is right: an earthquake can't change the axis of rotation, relative to a given inertial reference frame -- that is, the axis of rotation doesn't change relative to the "fixed stars" as a result of the earthquake. What the earthquake does is to move material around within the Earth, so that the position of the rotation axis relative to any given marker on Earth's surface changes.

If you prefer, rather than saying the earthquake shifted the Earth's axis, you can say that it shifted all of the stuff on the Earth's surface. Since we find it more convenient (often!) to use points of reference that are fixed to markers on Earth's surface, it looks to us like a shift in the axis.

Angular momentum doesn't change, but the angular velocity vector does. This is effectively due to a shift in the body's moment of inertia tensor.

There is some confusion about the term axis. The axis about which the Earth rotates of course did not change. It requires some external torque or transfer of angular momentum for that to happen. However, the Earthquake changed the shape of the Earth slightly, which changed the figure axis of the Earth.

Calculations also show the Japan quake should have shifted the position of Earth’s figure axis (the axis about which Earth’s mass is balanced) by about 17 centimeters (6.5 inches), towards 133 degrees east longitude. Earth’s figure axis should not be confused with its north-south axis; they are offset by about 10 meters (about 33 feet). This shift in Earth’s figure axis will cause Earth to wobble a bit differently as it rotates, but it will not cause a shift of Earth’s axis in space-only external forces such as the gravitational attraction of the sun, moon and planets can do that.

Source: http://www.jpl.nasa.gov/news/news.cfm?release=2011-080