Symbolism, Imagery, Allegory
It's not unusual for royal families to take some creature or other as a symbol for their family or clan. The U.K. has the lion and the chained unicorn, for example.
The same is true for the leaders in Stewart's book. Vortigern's forces are represented by the Saxon white dragon, while Ambrosius' forces are represented by the red dragon—the same red dragon you now see on the flag of Wales. These critters are emblems—symbolic representations of the leaders, their armies, and their major qualities.
The color of Ambrosius' victorious red dragon is also prophetic: Ambrosius beats Vortigern by burning him to death. Merlin's vision hints at this: "Shadows fled across the streams and staircases of fire, and the cave was full of eyes and wings and hammering hoofs and the scarlet rush of a great dragon stooping on his prey…" (III.10.43).
The redness of Ambrosius' dragon also helps Merlin out of a tight spot, when he sees the star called the "dragon of fire" in the sky above Vortigern's camp. When a gust of wind blows down the banner with the white dragon on it, Merlin can claim that the star stands for Ambrosius' red dragon triumphing over Vortigern's fallen white one: "King's Fort is not for you. You have seen the Red Dragon come tonight, and the White Dragon lie beneath him" (III.12.29).
Okay, so he's prophesying by the seat of his pants. But that's the beauty of emblems, which are visual metaphors: they give a lot of wiggle room for interpretation.