In theory, that’s true. In practice, the Director serves at the consent of the President and Congress, either of whom can replace them at their whim. There are no statutory conditions on the President’s authority to remove the FBI Director.
Robert Mueller served out his full term under Bush the Younger and two additional years under Obama (which required an act of Congress.) Before that, you have to go all the way back to J.Edgar Hoover to find one who did the full ten years, although William Webster almost made it-- he was moved to the CIA with less than a year left in his term as FBI Director.
