Think about it like a bucket. The water/atmosphere on earth is like a bucket the atmosphere is on top of the water. The moon is like a magnet that can attract the water and atmosphere. It can slosh around the stuff inside the bucket by moving around. But that sloshing isn’t enough to make anything leave the bucket.
The moon doesn’t actually orbit earth in thr sense one would think.
Both earth and the moon orbits their mutual barycenter.
This point is – because earth is alot bigger than the moon) inside earth. But not exactly at its center.
For this reason earth’s rotation wobbles a bit.
This is what’s causing the tides.
Its not the moon pulling at our atmosphere.
Here’s an animation that makes it easier to visualize.
https://en.m.wikipedia.org/wiki/Barycenter_(astronomy)#/media/File%3AOrbit3.gif
So the answer to your question is that the moon isn’t pulling away the atmosphere because the atmosphere is much more attracted to earth than to the moon.
The moon does not cause tides by lifting the ocean up; the moon causes tides by squeezing the Earth into a football shape (that’s why we have two high tides per day: there are two points on the football).
The atmosphere is also squeezed into a football shape, and so there are atmospheric tides as well.
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