This doesn't make any sense. The mass of an object clearly has an effect on how fast it falls towards another object due to gravity.

Gravity is a force acting between two objects. An object falling has gravity acting on it, but the body it's falling towards also has gravity acting on it and pulling it to the object (even if imperceptibly).

If I drop a lead ball on Earth, it will fall a lot faster than if I dropped it on the Moon. Consequently, the body it is falling towards (Earth/Moon) will also be affected differently by the object falling.

The "everything falls at the same rate" is simply a meme because we can't detect changes that small.