I think people make a lot of fuss about full moons because they think full moons are mystical.
The lunar phase only has to do with the angle of sunlight relevant to Earth falling on the Moon as it orbits. People who believe the moon's affect on the tides has similar effects on their bodies or psyche are thinking about distance and gravity. The light falling on the Moon has nothing to do with gravitational force. The light is a convenient measurement of time only, not matter. And it's only done from our particular vantage point in space. A person living on a planet at a different angle to the moon would see different phases, or none at all.
We only pick a particular phase and follow patterns based on that phase. Not all menstrual cycles correspond to the same phase of the moon, any more than more crimes committed. You can select any phase and begin calculating statistics from that phase and you will run into numbers that seem to coincide with it. People only relate more intimately to Full Moon or New Moon because they're more poetic than gibbous moons or crescents.