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Looking into the sky isn't the z axis. It's more like r from polar coordinates. Z in Cartesian coordinates goes through the polls of the moon and earth roughly.
From our point of view. Interpreting our field of view as the grid.
In that case the grid changes as you move on the surface of the earth. "Up" is relative to the center of the earth and is not the same direction in any two places. Cartesian coordinates (the usual x,y,z people think of with graphs) is difficult to use in this case.
I'm talking about staying in one spot.
The axis that the moon and earth rotate around is close to parallel. However, this axis is almost never directly "up" unless you are near the rotational poll. To visualize where this axis would be, you can look at the star Polaris or the northern star (check out a star finder app). To get a better sense of what is going on, perhaps look at a time lapse that shows how the stars move around Polaris.