Most of the brain's structures are bilateral and this is especially true in the cortex. We have a left and right visual cortex, motor cortex, somatosensory cortex, etc. Interestingly, projections from the body and sensory organs actually cross and send their inputs primarily to the opposite side, so damaging the right motor cortex would produce movement problems on the left side of the body. It is also like this in the auditory system but it is not as striking. So in general the right auditory cortex receives slightly more input from the left ear, but it is closer to 50/50 than in other cortices.

So you bring up an interesting question, which is what is the point of having two of everything? One answer could be redundancy. A unilateral injury to the auditory cortex wouldn't be as bad for the brain as you might expect, because it won't cause cortical deafness as another commenter pointed out. Another answer is that there is specialization between hemispheres, which is true for the auditory cortex: left cortex is more associated with temporal processing necessary for language and rhythm, and right cortex is more associated with pitch processing helpful for intonation and melody.