Nintendo attempted to do this to SMBX:
"On April 17, 2011, Redigit (Andrew Spinks) commented on his YouTube page that he received a call from Nintendo's lawyers asking him to remove Super Mario Bros. X from SuperMarioBrothers.org and delete all SMBX videos from his YouTube account. They also asked him to transfer SuperMarioBrothers.org over to Nintendo. He complied with their first two wishes, but did not want to lose SuperMarioBrothers.org. He said, "It is a good Mario fansite, even without Super Mario Bros. X." Nintendo was going to decide on that and get back to him. Redigit has asked that no one post copies of Super Mario Bros. X on the internet, as it goes against Nintendo's wishes and might make things more complicated for him." (
http://gaming.wikia.com/wiki/Super_Mario_Bros._X)
Five months later, Mr. Spinks made another announcement (which, unfortunately, I have failed to find) indicating that Nintendo has no legal rights to force the game to truly disband, for two reasons (note: this is American law):
1. The game is freeware and is not generating a profit for either Mr. Spinks or any direct SMBX affiliates (Episode designers e.t.c.) nor Nintendo or any of its affiliates, and
2. The game was not threatening to overtake legitimately licensed products of the Mario franchise in direct competition, i.e. SMBX is not more popular than any mainstream Mario game.
The second item is rarely known, but is very important because intellectual copyright holders can raise legal sanctions against those using their intellectual property,
even if they are not profiting, if the spin-off products/services are enjoying the same or greater success as the copyright holder's own products/services relating to the franchise in question. So, Nintendo could have only had SMBX sites taken down if they could prove the game was more popular then any of their officially released Mario games - since it obviously wasn't, Mr. Spinks was able to make a new SMBX site (
http://mariobrosx.webs.com/) without further harassment.
The idea behind the clause is that copyright laws are designed to prevent others from creating successful products/services which steal another's original ideas. Since "success" for a product/service is also defined in terms of popularity, a copyright holder could file a suite against someone using their idea without authorization even if that person was not directly profiting from any of it, providing the spin-off product/service was more popular - more "successful" - then the original inventor's works.
Obviously, Super Mario 63 is not more widely known then any of Nintendo's licensed offerings involving the Mario franchise, and since the game is distributed free of charge, Nintendo has no legal ability to have the game brought down.
Of course, legal threats are not illegal, which means that Nintendo can threaten sites like Newgrounds with lawsuits which would never come in order to coerce them to follow Nintendo's demands. In cases like this, they would comply to avoid any unnecessary risk. However, Nintendo has no legitimate ability to act on its threats if the game was not removed.
SM63 is no danger.