TTanguero nailed it regarding "weight sharing." It's not a "down force." It's a "horizontal force."
You would feel it not as someone hanging on you and pulling you down. Rather, you would feel it as as force that keeps you from falling forward.
Imagine holding yourself as far forward as you can, pretty much literally pushing your toes down to keep yourself from falling forward. Your partner is doing the same thing. and you are standing very close to each other, or even touching. (Close embrace!)
Now your partner relaxes and allows her upper body to "fall" towards you.
(This is best experiened without any contact with the arms.)
When she touches you, and you feel her weight on you, you can compensate by relaxing how much you push into the floor with your toes. You balance the force her body is exerting on you by allowing some of your weight to push against her.
Because men are in general taller and heavier than women, we say that the woman is leaning on the man. That's because it takes less lean forward for a taller, heavier person to put an equal "weight forward" (a horizontal force) than it does for a shorter, less heavy person.
Those words, "weight sharing," are not that helpful, but as far as I know, no one has come up with a better description. As others have written, "close embrace" usually has both partners "maintaining their own axis" independent of their partner.