I'm not too confident, but @Abhijit gave me an idea with his deck of cards thought:
The Joker in a deck of cards.
My neighbors are wacky!
One of them's a single-minded psycho who just likes cutting things up!
The ace in most cases is valued at one point, and as mentioned in his answer, performs cuts in the game of Bridge.
The other one's two-faced and doesn't like children.
Any face card in a standard deck (King, Queen, Jack) has two faces, and I guess they don't like kids because there aren't any represented in the deck.
They both have neighbors like them, but neither of them are like me.
There are multiple of each card in a standard deck, which is how it is referenced here as having neighbors like them. Admittedly, despite most games that use the Joker only need one in the deck, there are commonly two Jokers provided, so this doesn't quite work either. However, the thing each Joker has all to itself is a lack of suit. The face cards and aces both have many neighbors of the same suit, but the Joker - either one - has no suit.
I guess I'll just continue being I...
The joker, being a bit of a clown, is considered in royal stories to be having a mindset all his own, and would think of the rest of the world as being strange for not sharing in his quirky nature. This might also explain why he thinks his neighbors are wacky.
If this is correct, I'd like it to be appended to @Abhijit's post and give him best-answer.
Edit: Some ideas for myself when I wake up or anyone else who happens upon this: imaginary number i, arithmetic operators, or Iodine on the periodic table.