Only the the actual implementation code (and subsequent compiled result) is under copyright, anyone is free to write an independent implementation that's not based on any of the original copyrighted code.
(There is of course the possibility of patents covering the innovations in the SNES, but that's a separate issue from copyright, and I have no idea how software patents in Japan worked around the time the SNES got released.)
That's a good point. I believe the patents on the NES Lockout chip expired recently and hence the flood of NES/Famicom hardware clones in recent years.
Copyright is concerned with protecting original works, so it depends on what exactly you're referring to by "API". My interpretation is that the technical document describing an API is an original work and that the library implementing an API is an original work (provided they do not copy or modify existing works protected by copyright, then they are derivative works), but the idea the API conveys is not protected by copyright (this is what's known as the idea-expression divide[1]).
This is further complicated by the fact that we're discussing the law here so we can't assume anything to be logical or consistent. In the US, the DMCA have specific portions concerning the legality of reverse engineering for various purposes, which have implications for someone wishing to write an independent implementation of an API[2]. Maybe someone would argue that any implementation written according to the API specification is a derivative work?
I don't think there's a definitive answer to your question, since AFAIK this haven't been tested in court. In fact, in the ongoing patent/copyright dispute between Oracle and Google, Oracle claims Google violated Oracle's copyright by reproducing Java API's in Android[3].
(Disclaimer: IANAL, and I've probably misinterpreted something along the way)
(There is of course the possibility of patents covering the innovations in the SNES, but that's a separate issue from copyright, and I have no idea how software patents in Japan worked around the time the SNES got released.)