Informal & Playful:
* Sack
* Bag
* Pouchy
* Skin purse
* Little pouch
* Scrotum (for the male scrotum specifically, potentially offensive)
Descriptive & Neutral:
* Fold
* Pocket of skin
* Skin flap
* Skin fold
* Skin sac
Humorous:
* The old sack
* The family jewels pouch (for the male scrotum specifically, potentially offensive)
* The pouch of mystery
Medical:
* Skin pouch (used by medical professionals)
* Scrotum (for the male scrotum specifically)
* Labial folds (for the female vulva)
* Perineum (for the area between the anus and genitals)
The best nickname for a pouch of skin will depend on the specific context and the intended audience.
Remember: It's important to be mindful of the potential offensiveness of certain terms, especially when discussing sensitive anatomical areas.