As far as I know the Switch is the only modern console that supports local multiplayer. PlayStation and Xbox both require online for multiplayer. Nintendo likes to bring people closer to each other if at all possible.
I think it might get there, but I don't think it's accurate to say that it's the only console that supports local multiplayer. The PS4 and Xbox One both have games with local multiplayer. It's true that you need to pay to use their online multiplayer (you will also soon have to pay for online multiplayer with the Switch, although it's cheaper), but there are lots of games with local multiplayer and coop options. However, Nintendo has always been great at creating awesome local multiplayer experiences. MarioKart and 1-2-Switch are out now, and I'm sure others are on the way!
Well, I may have misunderstood the question... Are we talking about single machine multiplayer gaming? Or are we talking about multiplayer gaming with multiple machines? If we are discussing single system multiplayer gaming, like split screen or multiple players on the same screen, then Xbox definitely supports this without having to have Xbox Live Gold.