When I saw videos of the Wii Forecast Channel, I noticed the stars in the background were not really random, but seemed to resemble the rough pattern I'd expect to see in the actual sky, given dark enough viewing conditions. I couldn't confirm this until I got home and updated the channel to look around for myself.
The easiest constellation to recognize is probably Orion, so I looked for that one first. I vaguely recalled that Orion is a winter constellation which meant that I should probably look at the opposite side of the globe to find it in the background (at night). After a bit of shuffling around, I found it:
I noticed that the stars are indeed real star positions, relative magnitude, and even color! The top left star in Orion, Betelgeuse, is a red supergiant and even appears slightly red on screen.
The next obvious constellation to look for was the Big Dipper (Ursa Major):
It seems to be accurately positioned as well, which means that someone at Nintendo put in a fine bit of extra effort to include this subtle little bonus.