Do more than just colors- have visual representations that add to both the lore, the art style, and representation of color.
For example Legend of the Five Rings uses an incredible way to differentiate between "set of colors".
One of the most beautiful examples to take a lesson from. Take note the hundreds of ways they differentiate between the Clans. It's not just primary colors, but secondary and even tertiary colors as well. The colors are even included in hair style, type of wear, type of character, a symbol attached, flags, banners, mini-flags on the back, personalities.
The biggest thing I would take from this is the combination of multiple colors.
It is not a simple : Blue, Red, Purple, Brown, Yellow, Green, Teal.
It is a combination of two colors, with both being very pronounced. This can allow for a significant amount of additional number of players.
Each clan would have both a primary color, and a set of secondary colors.
"The Crab was mostly identified with blue-gray, plus black, red, and brown colors."
It is beautiful, because no matter what the person looked like (large, frail, samurai, wizard, peasant) you could always tell which clan they were from. It wasn't just "Red" for Scorpion, it was a dark Blood Red and BLACK. While Phoenix had sunburst colors: red, orange, yellow.
You would never confuse a Scorpion with a Phoenix, even if their colors were similar and they both used a lot of red.