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”
The ocean is not only very heavy, it’s also very long (tall, viewed sideways from POV of the Moon). This means the bits near the moon are more affected by gravity, pulled harder.
There’s so much fun to be had with this effect. When the moon is overhead, you’re stretched out an unmeasurably tiny bit, for example. It also causes tall objects to orient themselves perpendicular to moon (one end is lighter, the other end is heavier. Just like a weighted stick floats upright in water).
Oh, and then there’s the fact that your head (which could be seen as kinda momentarily orbiting the moon, with orbital period of once a month) has a higher orbital velocity than your feet, thus your head is constantly dragged sideways, and lying down helps alleviate this effect (only your nose is now pulled off your face, not your whole head). This is the reason why it’s easier to fall asleep when lying down.