The phrase "hunting for unicorns," or similar versions, is as old as the hills and has always been used in common parlance and literature to mean chasing after something we want that is mythical, imaginary, or unrealistic, and will be pretty much impossible to find. It is very similar to the phrase "tilting at windmills," which means fighting imaginary enemies. I am always surprised to find that this fact is not common knowledge.
It is NOT an "alternative lifestyle term." Come on, people. Just because it is often used in alternative circles does not mean that the phrase originated there. For example, in a biography of the poet William Carlos Williams, written in 1981, there is a chapter called "Hunting Down the Unicorn." That chapter is about Williams' efforts to complete a novel and the problems he had getting it published.
If you do a search of Google Books, you will find numerous passages in classical literature that refer somehow to chasing after or hunting unicorns. More examples: people looking to buy their dream home could say, "We're looking for our unicorn." Someone hoping to win the lottery could say "I'm looking for my unicorn" when they buy their ticket. I know I have posted about this before.
Now in poly circles, the phrase has come to be used most often to point out to partnered male-female couples how ridiculously unrealistic their fantasies are about finding a bisexual girlfriend to equally love them both. It is a response to pie-in-the-sky expectations - that is all! The phrase is not meant to say that triads, in and of themselves, are impossible. Nor does it mean that bisexual women who are into couples do not exist. No, it simply means that someone, most often a new-to-poly couple, has expectations that are so unrealistic to the point that they can never be more than a fantasy. "See here, newbie couple, you are chasing a unicorn if you think you can get everything on your wish list." It is NOT about triads, per se. It does not mean that a bi woman into couples is automatically a unicorn. Furthermore it is nonsensical for a woman to call herself a unicorn because that would mean she is a myth and not real.
Effectively, I (a straight woman) could say I am chasing a unicorn for wanting four boyfriends. It is a term that can be applied to many situations. So, let's just remember this, shall we? Calling people unicorn hunters simply means they are hoping for and pursuing an ideal which is a highly unrealistic fantasy. That is all.
/English lesson
P.S. And yes, UrbanDictionary is just like Wikipedia, which means it is user-contributed, so, not a reliable source. And I say this as someone who regularly volunteers to edit articles at Wikipedia.