Unique means one of a kind, without equal, &c.
But it does not necessarily mean that it shares nothing with anything else. The duckbilled platypus (sp?) may be a unique species, but that doesn’t mean that it has no characteristics in common with other species.
What word means that two or more things have no common elements?
Distinct?
Discrete? Mutually exclusive?
“totally different” may be as close as you can come in a general sense. If you have a more specific sense in mind, there may be a better term.
Antipode?
It’s not what you’re looking for, but I’ve been looking for an excuse to use it:
A word meaning two things that cannot exist at the same time, such as an immovable object and an irresistable force. The word is incompossible.
I found it in Ambrose Bierce’s Devil’s Dictionary, where he gives as the incompossible items God’s mercy to Man and the writrings of [someone Bierce didn’t like].
In set theory, disjoint means literally “having no elements in common,” but this is probably jargon.
Disparate might be close, depending on the context. It might need some qualifying adjectives if you’re using it formally.
Difform … although I commonly use the word dissimilar, or even wholly dissimilar.
unlike ? as in “unlike any other.”
“Dissimilar” is what came to my mind, as well.
(Bippy, Webster lists “unlike” as the definition of “dissimilar.” :))
So then that Sirius Cybernetics machine — which dispenses a substance almost, but not quite, entirely unlike tea — would instead dispense a substance “almost entirely dissimilar” to tea?
I prefer the original wording. 
“Duckbilled platypusses developed independent of other species, thus leading to a unique form, unshared by any other animal outside of Australia.”
Sometimes it seems better to use more words than the right word. Surely it exists, but no sense confusing the purveyor.
OT: CAN something have NOTHING in common with something else? the fact that they both exist as words in the english language is something they have in common among other things.
Incongruent?
Singular?
Something can have nothing in common with another object and not be singular itself.
Working on the lines of synonymous, homonymous and antonymous, heteronymous . And if it wasn’t a word before, it damn well is now.
Yeah—though I like being sent to the dictionary and the Golden Rule stipulates that I demand my readers be forced to the dictionary as well. :-p
The reason I’m asking is for a different sort of wording. Our township ordinance says that because Business Organization X is unique, it warrants a unique approval process. Someone is saying that because the process is unique, we can conclude that it shares nothing in common with other approval processes. So, I was wondering about a word for “shares nothing in common” so that I could juxtapose the two in the context of a larger discussion about the words themselves.
The platypus, for example, is unique, but it is a mammal and shares mamilian traits (I guess, I’m not a biologist).
BTW, thanks, everybody! I forgot to print this out to comb through dictionaries to find a favorite, so I’ll have to try to do that tonight.
Well, that doesn’t follow at all. Something can be unique in a single feature, and thus be unique itself, while still sharing all its other features with other things.
Nominalism?