For a reference point, here is the Switch sound effect.
After Va-Va-Voom! brought it up, I've been thinking about it and it bugged me because it sounded like something familiar. It hit me later that I had been listening to some of "Smashing....Live!" earlier today and it is at the 0:44 - 1:00 minute mark that has a sound effect similar to the Switch effect but not exact which you can hear with this video link:
When I snap my fingers, the sound from that snap is also close to the sound effect in the Switch video so it could be an altered finger snapping sound. The Nintendo Switch sound just sounds a bit more metallic with a larger echo though if recorded in a sound chamber my finger snap may sound different than the way it sounds in a normal room.
Ultimately, it wouldn't surprise me if the sound came from Nintendo's own sound effects. Perhaps they consulted with one of their in-house composers. After the Melee connection, I started listening to some of the sound effects from that game and there were a few snap sounds I came across that were close but less "popping" or "echo-y" than what I hear in the Switch video. It's fun to describe sounds! So, my best guess in the investigation right now is that it could be coming from a Nintendo game already and we just haven't made the connection yet.