An elemental spirit is an archetype of its given element. Some people believe they control aspects of a physical element, while others believe they are representative of the higher, symbolic aspects of the element(s) represented. There are also those who do not believe they exist, but that they are merely a symbolic representation, while others believe the element is more a representation of aspects of an entity tgan the other way around.
Just like with any entity or group, it is all a matter of belief, and not fact. The tradition(s) you study will have better answers than asking a very broad question.
A person does not 'have' an element. We are aspects of all the elements, representative on several plains from concept to creation to manifestation.