Tesla Inc.’s public feud with a top safety investigator is highly unusual in business, but classic Elon Musk.

The chief executive’s split with the National Transportation Safety Board this past week over its investigation into a Tesla car crash is the kind of brash move that Mr. Musk has made repeatedly in the 14 years since the electric-car maker was founded. From the kinds of cars he sells to how he makes them to how he promotes his company, Mr. Musk has made defiance of convention a point of pride, and made dismissing of...