Ah, I'm not saying there's a different force being applied to feather vs. hammer. The meme above doesn't mean that they "fall faster" in the sense that the hammer falls at a higher velocity. It's rather colloquial usage of "faster" to mean "finishes sooner". Because what does happen, is that the hammer collides sooner with Earth, since the hammer pulls the Earth towards itself ever-so-slightly stronger than the feather does.

I guess, for this to work, you cannot drop hammer and feather at the same time in the same place, since they would both pull Earth towards themselves with a combined force. You need to drop them one after another for the stronger pull of the hammer to have an effect.

So, this is also going off of this formula:

F = G * mass_1 * mass_2 / distance²

But setting mass_1 as Earth's mass and mass_2 as either the feather's or hammer's mass. A higher mass_2 ultimately leads to a higher force of attraction F.