Is it possible to get a duplicate legendary from the Frozen Throne prologue?
No this can't happen.
This is a reddit comment from a Team5 Member:
Not to worry! Non-pack rewards, such as those offered from the Welcome Bundle, Arena, or Heroic Tavern Brawl will follow the same duplication rules that will apply to opening card packs. The same will be true for the upcoming Prologue reward. You won't need to obtain them in a specific order to avoid getting a duplicate Legendary.