Many bodily organs are formed in pairs e.g. arms, eyes, the brain left, right hemispheres. Is there any difference(s) between each of them, in any way(s).?
Our two eyes work complimentary to each other in order to allow our brains (predatory animals that we are) to perceive, narrower yet, 3D vision. Only half the optic nerve fibers chiasm across the opposite hemisphere, something that also may contribute to the stability of the brain but also allows for slightly different images to be combined.
You may find this interesting (even if it is journalistic Science): http://goo.gl/et5sGO
I think the brain is the key to your question: no, the two hemispheres are not identical in function. Not only is one cortex dominant but different attributes have been mapped to homologue regions of the two hemispheres.
Similarly, most bodily parts that come in pairs are not identical. Even our feet are not of exactly the same length or shoe-size, due to favoring one over the other.
Keep in mind the use-it-or-loose-it principle. People with drastic brain surgery or severe birth challenges, through use, have rewired their brain to compensate. Loosing the right arm will cause someone's left one to increase in dexterity and strength after some time of use. The same is true for some cases of stroke victims, when the function that used to be performed by the anoxia-damaged regions are soon picked up by a different region and, in some cases, functionality is, to an extent, restored (Frost et al, 2003).
Not all regions are equal though. Having one hippocampus removed will not cause global amnesia because the contralateral one will pick up (Gennnaro et al, 2006). Remove both of them, unfortunately, and anterograde amnesia sets in (no new memories can be engrammed - remember the infamous case of Henry Molaison).
Good reference for Dimitrios points at: http://www.ncbi.nlm.nih.gov/books/NBK10953/
As an aside: one thing that comes into common culture as a result of OVER emphasizing the right/left brain dominance is the Right brain=creative thinker and Left brain= logical. This tradition might not be the case as a recent study mentioned in this article by University of Utah.
See this at http://www.dailymail.co.uk/sciencetech/article-2395638/Are-right-brained-creative-left-brained-logical-Youre-claims-study-link-brain-sides-personality-nonsense.html
"During the course of the study, researchers analysed resting MRI brain scans of 1,011 people between the ages of seven and 29. By monitoring brain activity, the scientists were able to correlate brain activity in one region of the brain compared to another.In each person, they studied functional lateralisation of the brain, measured 7,000 brain regions and examined which regions of the brain were more lateralised. They looked for connections - or all of the possible combinations of brain regions - and added up the number of connections for each brain region that was left-lateralised or right-lateralised. They found no evidence that individuals use their left-brain network or right-brain network more often."
There are anatomical differences between the left and right lungs. For instance, the right lung has three lobes whereas the left lung has two. The right lung is typically shorter and heavier than the left lung.