The Earth's interior is not a rigid body. Its innermost core is, and its crust is, but the mantle and outer core are liquid. The axes of rotation of all layers of the Earth are basically aligned, not perpendicular.
This is Earth, not Uranus.
In fact, the inner core is theorized to be rotating at 5 times the speed of the crust.
It is a nearly constant rotation but there is some subtle slow down due to tidal torque interactions with the sun and moon.
Even if it were constant, that isn't what physicists call a "perpetual motion machine". Nothing in Physics forbids a body to continue moving forever. "Perpetual motion machine" refers to a machine that will move forever, but that we EXPECT to be able to extract energy for our own use and not affect said machine. It really should be called a "perpetual work machine". Whatever it is called, it violates the laws of thermodynamics as we know them.
Earth's "axis" is represented by an imaginary line running between its physical north and south poles around which the planet rotates. That line is *generally* but not exactly perpendicular to Earth's axis of rotation.
"...would the Earth's core spin at a constant velocity?..."
Because the *inner* and *outer* cores are not rigidly connected to Earth's solid mantle, the possibility that they rotate slightly faster or slower than the rest of Earth has long been suspected but so far not definitely confirmed.