The Burgundians in the game and the original, Germanic Burgundians are two different groups.
This is more easily understood in French, with both groups having distinct names: Bourguignons for the 100-year war era civ (and used for the playable civ), and Burgondes for the germanic people. Of course, the Burgondes gave their name to Burgundy (Bourgogne), which the Bourguignons got their name from.
This is actually kind of the same relation between the Franks (Germanic people) and the French which derived their name from them while keeping their Latin identity.
Also I think a lot of people missed the real reason why the Burgundians were added in a DLC: I think it just comes down to a wink to the original game, where the Burgundians played an infamous role in the Joan of Arc campaign and even had an appearance in Barbarossa. Dukes of the West also added an Edward Longshanks campaign, and a campaign showing the Burgundian side of the 100-year war.