Why is my iphone 6s 64gb rosegold color gets loss due to manufacturing fault?

If it were me, yes, because it's not cosmetic damage (the proof is in the photos) and it's a special circumstance. I've never heard of anyone getting an allergic reaction to a cellphone, but I'm sure it's in the very tiny minority.

After you get tested, and find out what's wrong, then go to Apple and explain it to them.

Uh what did you do? Is that the paint on your hands, or the finish?

First off, since it is an allergy to your skin, get that checked out. This could be a sign of a more serious medical issue.

Second, keep it in the case until you get your skin tested.

Third you might want to go to Apple (if available) after the first two have been done. If you have to handle the phone out of the case, use gloves. Explain the situation to them. Whatever happens with Apple, I don't know. If it was me, and I was with Apple, even though cosmetic damage is not covered, I would exchange it, because it's a medical issue and there is no damage that was user caused. I would go in with a doctors letter explaining this. They may not exchange it, but then again.

Fourth, do not take it out of the case unless you have to.

I would highly suggest to get that checked out by a doctor. It looks like it's damaged, but since it's caused by a skin allergy, I would get that checked out.