Ah! Hojo as a Madman was my next guess, but I didn't even consider Red XIII - I was thinking his Bad Rap or his HO512, but Red XIII makes so much more sense - he's red, after all.

So, here we go again. I was trying to crack this while watching Seinfeld - in other words, I probably failed.

The blood of the planet, (Lifestream)
That runs over mountains, (Green or Black Chocobo, or it refers to line 1)
It has halitosis, (HO512 does a breath attack)
But it can do enough spells to make up for it. (It also casts some stuff, I guess)

Each stands alone not with the rest, (the lines of the original riddle refer to separate things)
Finding the link is what I test. (find the commonality behind them)
A madman named an unlucky beast, (Hojo naming Red XIII)
And aptly so to say the least. (Red XIII is red)

Now I'm having trouble with fitting in lines 1/2 with the others...
- Lifestream
- HO512
- Red XIII

Then again, Red XIII is to be used as a clue to what lines 1-4 should represent, and is not part of the riddle itself...

Hell, maybe the link is Red XIII. The Lifestream embodies itself in materia that Red uses, and he and HO512 were Hojo specimens. So I'll give that as an answer.