What is the chemical structure of the cocoate ion vs. the olivate ion?
Note 1. Both coconut oil and olive oil are natural oils. The soap, produced from them, is mixture of several compounds.

The difference of olive and coconut oil may be easily found in wikipedia: http://en.wikipedia.org/wiki/Coconut_oil#Composition_and_comparison 

In short, coconut oil contains mostly saturated acids esters, while olive contains mostly unsaturated ones. The unsaturated acid esters are responsible for olive oil ability to solidify on air. But here is 

Note 2: In my knowledge, allergy is usually caused by carbohydrates or proteins. Fatty acids and their ions have too few binding sites to be recognized by immune system as something distinct from our own. The real source of reaction may be 


Minor components of initial oil. There are quite many of them and removing them completely is not easy. 
Other components of soap, such as perfume or pigment.
Soap solution pH, that may differs from soap kind to kind, but this is unlikely.