I notice that in many cases, the Wii name stands out on its own. It's the Nintendo Wii, sure, but it most cases it's just "Wii". Even on the
official Wii site the name "Wii" stands alone pretty often, including in the upper-left corner.
Remember the mid-eighties? People said they "played Nintendo". It was pretty much the way to say you were playing video games (though maybe because at their peak they had like 90% of the game market so there was a 90% chance that you were playing on the NES anyway).
People also said they were "playing Sega" but no one ever says they're "playing Sony" or "playing Microsoft". Maybe because Sony and Microsoft have other products but Nintendo is still nearly entirely games so I see no harm in still saying that someone is "playing Nintendo".
If Nintendo's not reffering to it with "Nintendo" in its name, do they expect us to be saying "I'm playing with my Wii" / "I'm playing Wii" / "I'm playing Wii games"?
I remember having to write an apology in Grade 1 (that's "first grade" to you Americans) for overusage of the word "poop". I bet some kids are going to get in trouble in elementary for saying "I'm going home to play with my Wii" even if they weren't even trying to be funny by being "innuendish".
Come to think of it, I'm not sure why I'm thinking about this now. It's not like the name is new. Maybe I just missed it when people said "playing Nintendo" and it only just dawned on me that even Nintendo doesn't want that now.