The Wii Classic Controller is wired. The port on the front of this console is the same port on the bottom of a Wii Remote, which makes it pseudo-wireless in a way, but there's no Wii Remote in this case so it is just wired.
Yup, it's I2C with a 1394-ish connector (which makers have taken advantage of before), so it is very easy to implement... with the added side benefit of selling a few of the standalone controllers for Wii[U] virtual console use.
The console itself is probably a simple ARM chip - the BCM2835 in the first-gen Raspberry Pi would work perfectly, among tons of other possible SoC's. It will probably be under a plastic blob, but someone will find out sooner or later ;)
The Wii Classic Controller is most certainly wired, just it's wired to the wireless Wii Remote. You may be thinking of the Wii U Pro Controller which is fully wireless.
I wouldn't mind if it were just the fact that the Wii Classic Controller is wired. What bugs me is that the cable is really, really short: only long enough to run into something that you'd be expected to hold in your lap. Of course, that's exactly the use case it was designed for, so it made sense in that context. But it causes problems when you then want to plug it into something several feet away, like, say, the console.