Even though Nintendo has said it the Wii is capable, they are hodling back on adding that functionality just yet. Shigeru Miyamoto says on the issue:

Certainly as we've been working on Wii U, I think we're at a point where we don't feel it's realistic to expect people to purchase two GamePads yet. We're still trying to get as many people as possible to buy the system itself, and that's where our focus is right now. For the time being, two GamePad gameplay is not within our objectives in the short-term.

From a technical standpoint, I think if we decided to pursue that, technically we would be able to and we can perform system updates that would allow for that. It's also very interesting from a gameplay perspective idea because there are a lot of possibilities with two GamePads. But, at the same time, taking that kind of approach would again be a drain on resources and require us to continue to utilize our development resources for that. So then it would become a question of where do we want to devote our resources: Is it to creating regular GamePad games, or creating games centered around two GamePads?

Certainly, the added gamepad is the last thing on people's minds. The Wii U's still in a precarious position in the market, even if it's gained considerable ground/momentum in the past few months. Nintendo's focus is still on defining the use of the single gamepad the Wii U has.

Sources: NintendoLife and IGN