I think the reason some people might believe this claim is because we’re taught in school that the moon’s gravity causes the tides. I think the reasoning goes, “well if the moon’s gravity can affect the tides, surely it can affect smaller things too”
No. If gravity on such a small scale affected us, we’d have evolved to incorporate it as many animals do for navigate, or evolve to ignore it. We did the latter, if it was ever necessary. The gravity of the earth is many magnitudes greater. And you can prove that the shifting gravity from the moon is moot by the sheer fact that you can sleep on your back, either side, or stomach and not feel inverse affects. You can even move around through the night, like the moon does.
Also the tides thing is half right. About half as strong in tidal energy is the sun. Solar tides exist. So the question would also need to be “does the sun’s gravity affect us?” And the answer is the same as before.