When Palmolive made that claim, it was comparing its product to past cleaning agents. I was curious about what people used to use, so I looked it up. Seems that abrasives (sand, plant material, ashes) were common, and soap. Like, OG soap, made by combining oils with a chemical base.
Interestingly, the ingredients to Dawn dish detergent do not include soap. It contains surfactants (chemicals which break surface tension and in effect make water “wetter”), and detergents. Detergents emulsify oil in water. Water molecules are polar, they have a positive charge at one end, and negative at the other. Fats and oils are non-polar, which is why they don’t mix with water. Detergents are molecules with a polar component attached to a non-polar component. They attach to and surround non-polar molecules, and make them polar/hydrophilic/able to dissolve in water.
Dish detergent will still remove oils from your skin, but at least it’s a lot better than abrasives and corrosive chemicals.