Tiger skin needs to be analyzed taking into account Nataraja and Durga.
Also every single Hindu diety, Male or Female is paired with an animal. You are partially correct but it's a bit more.
The texts describe the animals more as vehicles. https://www.learnreligions.com/vehicles-of-the-gods-1770297
That which carries, moves or pulls.
Yes it represents, the lower desires and motivations. In some context it can be closer to the idea of the Jungian shadow, but only if it's repressed, unconscious or unintegrated.
For the Durga - Shiva parallel, one represents Shakti, energy, movement, action. The other represents contemplation, reflection, meditation and perfection. There is also the form of the two combined, union of male and female energies, Nataraja.
Durga rides the tiger, it being under her control. For the slaying of demons, for going into battle, she is perhaps the most powerful of all the Gods.
Shiva sits on the defeated, destroyed tiger skin.
There is a couple of lessons here. Don't kill or attempt to kill your lower desires. Control them and use them as best you can to move rightfully throughout the world. If you kill, main, subdue your animal half, your maiming yourself and halving your own power.
You will always have that lower half of yourself, and it will always be there, except if you take the path of the meditating yogi.
That being said, Krishna said that the path of action is superior to the path of inaction.