Historically, Dragons only accept one rider. And no person can ride more than one dragon in the same time frame. In his histories Martin explains this explicitly. It is the reason why Aegon the Conqueror only ever rode one of his three dragons. His sister-wives each rode one of the others.
The exact reason why this is true is unknown. But it is also unknown why anyone can ride a dragon at all. "Scientifically" it might be explained by pheromones and the territorial nature of dragons. But, given the flavor of Martin's world, it probably has more to do with the mystical connection between the man and beast. We know that such mystical connections are possible from the mechanisms of the various wargs in the books and show.
As for why Jon was able to make friends with Drogon, that likely has to do with his parentage. It also may have been an unconscious manifestation of innate warg capabilities. But not being eaten by a dragon is not the same as riding it. Daenerys's other two dragons like her and are friendly towards her, but she cannot ride them.
So the million dollar question for Game of Thrones is who will ride Daenerys's other two dragons.