I took it for granted that, since it was impervious to weapons, Herakles must have worn it as armour. Herakles kills the Nemean lion and wears its hide, which suggests that he becomes the lion or beast. I agree that it manifests his animal nature, but there must be some reason for wearing a lion skin, even on a hot summer day.
All good warriors have some kind of armor or shield.
It's like Athene with her aegis or Achilles with his mom's armor.