In case there was any doubt, Zelda producer Eiji Aonuma has revealed that The Legend of Zelda: Breath of the Wild will be single player only. However he has stated an interest in doing another multiplayer Zelda, perhaps as even a tie in to a generally focused single player Zelda.
While speaking to IGN about the subject of multiplayer in the future of Zelda he shared:
“I would like to take what I learned from Breath of the Wild and see if we can somehow fuse those learning points into another multiplayer Zelda. For example, with Triforce Heroes, which followed a similar format of Four Swords, there was a multiplayer involved in that game. That’s definitely a possibility and we will continue to [experiment] throughout the Zelda franchise.”
Eiji Aonuma has not been shy about wanting to push the typical conventions of the Zelda franchise, and when Breath of the Wild was first officially revealed he shared that he wanted to push conventions in the game. More notably, he wanted to change the way we looked at the Zelda experience with other people, many translations of this suggested he wanted a major 3D Zelda game to be multiplayer playable, but I have a different translation. Breath of the Wild is about the personal story, and the fact that everyone who plays will have a different experience with the game, this is akin to games like Fallout which the players decide their own journey and experience.
In terms of multiplayer, I don't think this idea really fits Zelda as proven by Tri Force Heroes, and even Four Swords, but who knows maybe they could possibly surprise us.