It is obvious that the rotation of any planet is a combination of rotation along three axis x,y,z.
You happened to choose a frame of reference, the ecliptic frame, in which the Earth's rotation axis isn't oriented along one of the reference frame's principal axes. You wouldn't see this as a problem had you picked a frame in which the fundamental plane is the Earth's equator rather than the Earth's orbital plane.
Rotation in two dimensional space is tame, quite weird in four dimensional space, and weirder yet in even higher dimensional spaces. Rotation in three dimensional space (e.g., our universe) lies somewhere between tame and weird. One consequence of rotation in three dimensional space is that a rotating object can be described as having a single instantaneous axis of rotation. The is one of the many consequences of Euler's rotation theorem (referenced in CR Drost's answer).
From the perspective of an equatorial-based frame, the Earth's rotation is almost purely about the z axis. There are a number of factors that make this "almost purely" rather than "purely". A rotating rigid body with non-coinciding angular velocity and angular momentum vectors will undergo a torque-free precession. While the Earth isn't a rigid body, this torque-free precession does exist, with non-rigid complications. This is the Chandler wobble.
Another factor is that the Earth comprises multiple rotating elements: The mantle+crust, the outer core, inner core, atmosphere, and oceans. There are seasonal variations in the Earth's rotation rate and its rotation axis due to different behaviors of the atmosphere in northern hemisphere winter versus summer, transfer of water (as snow) from equatorial regions to Siberia during the winter and transfer of water (as melted snow) back to equatorial regions during summer, and transfer of water due to complex ocean currents. Exchanges between the mantle and core result in decadal variations in both the rotation rate and the direction of the rotation axis.
Even longer term, torques from the Moon and the Sun make the Earth undergo a slow but large precession of roughly 26,000 years. This axial precession is one of the key causes of ice ages. Even longer term, tidal forces from the Moon and the Sun are making the Earth's rotation rate slow down, and most likely are subtly changing the direction of the Earth's rotation axis.