Grenouille is born with a supernaturally developed sense of smell. He can smell the approach of a thunderstorm when there's not a cloud in the sky...

It makes a lot of sense that Suskind would choose the sense of smell as the medium that shows the supernatural origin of his main character. 

From a biological perspective, people are innately born with a uniquely particular smell. We produce pheromones, and the scent of these is recognized by our pets , our babies, and we can even detect it in each other, at a subconscious level. It is the scent that makes us human. Grenouille was born without it. This made him not only inhuman, but also the subject of much speculation since his birth.

Impossible! It is absolutely impossible for an infant to be possessed by the devil. An infant is not yet a human being; it is a prehuman being and does not yet possess a fully developed soul.

By not giving Grenouille a human scent, Suskind takes away his "humanity" while, ironically, forcing Grenouille to search for it in the scent of others. Tragically, Grenouille's ability to detect smells is superhuman, which adds insult to the injury. The very trait that makes Grenouille unique will end up being his tragic flaw. 

From an artistic perspective, notice that smell is the only one of the senses that you cannot recreate in paintings, music, plays, or literature. You can speak about it, describe it, and perhaps even remember it. However, the memories and emotions that surface from specific smells are impossible to recreate with art. You can look at a picture and tell everything about it except the unique scents that were present at the scene.  Similarly, Grenouille is so strange, unique, and enigmatic that he cannot be understood enough to be able to fully describe him. Nothing and nobody is like him. 

From a historical perspective, it is arguable that 18th century France was a pretty unsanitary place

In the period of which we speak, there reigned in the cities a stench barely conceivable to us modern men and women. The streets stank of manure, the courtyards of urine, the stairwells stank of moldering wood and rat droppings...

For someone who is described as "gifted and abominable," what better punishment is there than to have an extremely sensitive sense of smell in a city rife with putrefaction? Moreover, Grenouille is disgusted with the inner smell of humans, namely, with the smell of humanity. He distances himself from society for a time for this reason, perhaps sparing hundreds from his unstable nature. 

Now, your other questions ask whether this gift is possible. Hyperosmia is a rare disorder that is very similar to Grenouille's sensitivity to smells. However, this disease is linked to other neurological conditions and in no way would it make anyone able to discern supernatural smells, such as those Grenouille was able to detect. 

Lastly, it is greatly possible that Suskind wanted this novel to read like an allegory, showing us that life is not limited to breathing and using our 5 senses. We are all complex and unique in our own ways. Still, there is no question that, at the heart of the novel, the theme of humanity is what propels the situations that take place: Grenouille's search for something that only he can understand is no different than our everyday search for something that will give us purpose and make our lives worth living. Of course, his search ends up in a tragedy and murder is a capital sin. Murder goes completely against humanity, too. This is what differentiages Grenouille's search for meaning from ours. That is what makes him "abominable."