This comes out of the French New Wave and the emergence of the film auteur.
I think directors like David Lean, Orson Welles, John Ford, Akira Kurasawa, Viktor Fleming, Alfred Hitchock, Sergei Eisenstein and a few hundred others like them had a "directing style" before the French New Wave.
A director's style, to me, is just whatever makes a film unique to that director. In the case of someone like Tim Burton, its the choice of material and especially art direction of his earlier work was unmistakable and unique to Tim Burton.
For someone like Stanley Kubrick, it didn't matter what genre he was in, his style was in his handling of the story in terms of pacing, lighting, cinematography and editing that make his movies unique to Kubrick. For Steven Spielberg there is almost always a story about a father not being the best dad as a recurring theme in the material, as well as whatever visual he's showing causing the actors to give a non-verbal, slack jawed look of "awe" at whatever it is (dinosaurs, aliens, a mothership, etc.).
Wes Anderson's style is to not use the rule of 3rds that much have have people framed dead center. John Woo always has someone jumping sideways while shooting one pistol in each hand (usually a .45).
So in order for a director to have a "style" of directing, they either have to have a body of work with some kind of thematic or visual consistency, or even if they've only ever made one film there is something intrinsically specific to that director about it.
It's not an easy thing to define. It's a very subjective thing, a "director's style".