You are correct in saying that this is just a fictional story, but as with many stories, the mythological sources shine through. With the case of Tolkein, the myths and histories he is using is Germanic.

Now, the matter of the ring at hand. There were other rings, and each ring fulfilled particular desires, but the one that Saruon wore, ruled all of these other rings, which means that it fulfilled all of these desires or even needs in some cases.

The other rings each fulfilled the desires of the wearer. Tom Bombadil wasn't affected by the ring because he had all he could desire. Gandalf already posessed a ring, and his thinking would probably be that by fulfilling all desires, he would have to stop wandering. After all, we wander fo fulfill certain desires, and if these desires were fulfilled, a person would have to stop wandering. The rings also represented different desires for the culture. The Elves biggest desire was fertility. The Dwarves, currency; the humans, self-sufficiency.

I think the fact that this one ring could rule them all is interesting. It leaves many possibilities open to the interpreter.