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”
Is it time that’s allowed the moon’s gravity to affect the tides? I think it would cause tides even if it suddenly appeared out of nowhere into orbit.
Now, I’d need someone else to explain why the effect on the tides is so visible. I’d suggest that it is because the water in the oceans both has a huge mass and is fluid, so its shape can change easily. There’s a difference between the moon pulling a portion of the water closer to it on one side of the planet and the moon “stretching” a person who is caught between the Earth’s and moon’s gravities.
The bulge on the opposite side of the Earth from the moon is because the Earth is being pulled away from the water.