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”
For billions of years. There was once a time when tides were massive and the moon took up most of the night sky (it’s slowly being slingshoted out of orbit and will eventually become a dwarf planet orbiting between earth and mars, but the sun will likely engulf the earth system before that happens)