Director David Lowery explained why Elliott is a furry dragon in an IGN article, saying that he'd rather have "the kind of dragon you really want to give a hug to" than a Game of Thrones (2011) type dragon, which he described as "cool, but scaly and cold".
The film takes place in 1977 and 1983. The original Pete's Dragon film happens to have been released in 1977.
Levi Alexander, who plays Young Pete in the prologue, and Gareth Reeves, who plays his father, are actually related. They are first cousins, once removed.
Director David Lowery described this film as a "re-invention", rather than a remake, of Peter et Elliott le dragon (1977). He wanted the film to distinguish itself from the 1977 film as much as possible, and reinvent the "the core story of a venerable Disney family film". Lowery even watched L'Étalon noir (1979), Yume to kyôki no ôkoku (2013), Mon voisin Totoro (1988), Le Voyage de Chihiro (2001), and The Witch (2015) for artistic and visual inspiration.