This is what I think about their zanpakutohs:

The sword is made up of a person's spirit power. Their power is compressed into the general form of a sword. I guess you can say your spirit power is another being inside of you, so that's why the sword has a name. The true form of your sword is the actual form of it, it's just that for convenience sake, the sword is kept in that general sword/katana form. As for having two swords, they may combine to make the sword's true form, such as Ikkaku's spear, which used his sword and his (guard? not sure what its called, whatever you keep your sword sheathed in) to make. Not all people can have a sword, since they have to have 1) enough spirit power to make a sword, and 2) enough control over it. One always has spirit power, it's just if he can actively awaken/release it at will and be able to have control over it.

Well, at least that's how I think about it.