It does. The Earth’s tilt is not a constant angle relative to the orbital plane. Not only does the magnitude of the angle change but so does its direction. The equator is not defined relative to the rotation axis except approximately. Having the coordinates of the Eiffel Tower change because of axial drift would be annoying.
Imagine a cone with the point sitting on the ground and its base parallel to the ground. That cone represents the Earths precession. The time of the represents the point where the earth intersects the plane of the ecliptic, the rim of the cone represents the possible direction the earth will point as it precesses. But no mater which direction the pole is facing the angle between the earths axis and the plane of the ecliptic remains (roughly) the same.
Basically at the same the Earth is rotating around its axis, rotating around the point of precession AND orbiting the sun on the plane of the ecliptic.
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