I am familiar with some of the definitions of crystal momentum and I am familiar with how it is related to Bloch's theorem. I also am familiar that crystal momentum is not the momentum of each electron. However, does crystal momentum have a physical meaning? Is it an observable quantity? Whose momentum is it? Electrons, phonons, or something else?
Crystal momentum is a modulo momentum. As said in the comments, momentum is somewhat difficult to observe, as we usually measure velocity and multiply by mass.
A good way to "visualize" crystal momentum is watching how spokes on a tire appear to precess when observed under 60 Hz streetlights. You could think of your wheel's "crystal momentum" by observing the apparent angular velocity and multiplying by the wheel's moment of inertia. The wheel is likely spinning much faster than it appears when under a strobe, but because of the period nature of its spokes, you only see the modulo velocity.
This effect shows up in crystals, where each crystal site looks almost the same, and your wave vector is "sampling" every spatial period.
I would only add to Jonathan's answer a picture (which I can't put as a comment):
(from Kittel, Solid State Physics)
In this image, think of the state as completely determined by the value of the wave at the solid points (which are like nuclei). Then it is clear that both waves, despite having different wavelengths, represent the same state. This is really a practical application of sampling theory.
Alternatively, if an electron in a crystal is prepared in a momentum eigenstate that is like one of these two waves, the crystal will scatter it into states like the other one, as well as other wavelengths that belong to this "class" of being the same at every black point. So the full momentum is not conserved, but this sort of "discrete momentum," which is the values of the waves at these points, is. And that's crystal momentum.
In a real crystal the nuclei are not points, and have an extended force, but as long as the potential is periodic this logic turns out to still work, as proved in Bloch's theorem.