Crab - Big Beatsticks and Defensive traits. Though they did have a neat period where their schtick was a trait that triggered on their death that potentially let them take opponents down with them.
Crane - Honor running, dueling, and stalling. These were the core playstyles for Crane through much of their history.
Dragon - Enlightment and Dueling. The Dragon had the most consistent Enlightenment decks over the years. They also had a 2 turn win deck that was in Standard Rotation for awhile, but it required like 6 fate cards and a boxable personality to show up in your starting hand, I actually had it pop against me once.
Lion - Tacticians and Followers. The Lion also had a lot of built in force bonuses. Their high median personal honor allowed them to also run good Honor/Military switch decks where they'd blitz early and then sit and honor out.
Mantis - Ranged Attacks and Economic Control. They also had a strong Magistrate theme for much of their history.
Phoenix - Shugenja and magic tricks. I had seen a few Phoenix enlightenment decks but they were way rarer than Dragon ones.
Scorpion - Dishonor and out of battle P-Kill. Ninja tricks and dishonor on you and your cow. It was funny watching high end Honor and Dishonor decks slap fight each other and hover near starting honor.
Shadowlands - Big Beatsticks and Fear. They also had a bit of dead personality manipulation. I did run a funny alt win condition Shadowlands deck back in Jade based around Master of the Five and the Elemental Terrors. I even pulled it off a couple times at kotei. I never made it out of swiss with it though.
Unicorn - Cavalry, Switch, and Tacticians. The ponies were the classic Honor/Military switch hitters. They were my favorite for most the time I played. Their superior economy also let them run more of the better items. Their tacticians weren't as good as Lion ones but were about as numerous.