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 moon would affect the tides regardless of how long it’s been there. Calculating the effect of the moon’s gravity on the earth is a standard exercise in a graduate classical physics course. In essence, the moon’s gravity “squashes” the earth at the two poles, causing tides at the nearest and farthest end of the earth.