Everyone starts R&D for the next console soon after they release one. What's interesting about his quote is the "we’ve also said publicly" part.
Even though Sony began R&D for PS4 right after they released PS3 in 2006, they did not publically say it existed until 2013.
Reggie is saying that they have publically announced that they are working on their next home console. And the only public announcement I know of that fits this, is the NX announcement.
Further more, Metroid's producer Tanabe said this: "If we started [Metroid] for Wii U now, it would likely take three years or so. So it would likely now be on Nintendo's NX console,".
He calls the NX a console, and more importantly, said that a home console WiiU Metroid would likely end up on the NX console instead. That certainly suggests that NX is a home console, as it doesn't make sense to move development of a home console game to something else. Especially without explaining why, since this was an answer to the fans, and it would be impossible for them to discern that he is refering to something other than a home console from his reply.
People can specuate in different ways, but to say that nothing suggests it's a home console is simply not true.
You're using the term "confirmed" in the end, and I agree that no one from Nintendo has confirmed this. But there's certainly comments that suggest it.