The cinematic is just made to look "realistic" (or at least somewhat plausible) and no ruler of nation would step on a battlefield in cloth "armor". This has nothing to do with classes, neither mechanically nor in lore. The concept of classes simply does not exist in lore as it does for us players. There is no magical limitation on swords that prevents a priest from picking them up, technically anduin wouldn't even count as a proper priest since he probably was never ordained. He is just a prince that read some stuff and received a formal royal education.
Drakka can carry around 2 daggers on her belt and 2 swords and use what ever she seens fight depending on the situation. While that might represent multi-classing as in games like D&D, at the end of the day it means nothing in the context of WoW.