I checked the book for Mario Paint and the SNES Mario Kart; I didn't see anything obvious one way or the other on this topic. However, systems more recent than that (N64, gamecube and wii) definitely DO have something to say. It actually says the exact opposite of what you suggest:
"Copying of any Nintendo game is illegal and is strictly prohibited by domestic and international intellectual property laws. "Back-up" or "archival" copies are not authorized and are not necessary to protect your software. Violators will be prosecuted."
-Inside front cover of Smash Brothers: Brawl for wii.
They actually manage to make the message even more infuriating by telling outright lies. (Not necessary? Are they seriously implying that their disks can't ever be scratched by anything? Or that an N64 cart can't be killed by ESD?)
So, while they may have once been cool about it, at some point they decided that being jerks was the way to go. It's been this way since N64, so I'm sure the DS games say the same thing.
Note: I looked in the book for Mario Kart 64, and found this exact message there, too. It's possible that third party titles don't have a message this ridiculous. I remember it being in the gamecube manuals, too, but don't have one at hand to check for sure.