Nintendo was for decades one of the biggest videogame publisher in the world (still is), of course they consider rivals the other software publishers.The difference is that the new management isn't antagonizing third-parties or sounding arrogant the way they were back when. Yes, Iwata was antagonistic toward third-parties with quotes like
I'm glad Kimishima is avoiding doing statements and decisions like this. [so far]
People should realize by this point that Nintendo isn't Sony or Microsoft which are for the most part content with controlling the platform, Nintendo want to be the top publisher in the world.
It's not a coincidence that almost for every Nintendo system, 30% or more of the software sold was produced by Nintendo:
I see no change in the relationship with third-party, what I see is that for the first time Nintendo seem to be using an architecture/API supported by the most popular engine and thus is likely to receive good support from third-party (hoping that Switch will sell good).
For example I expect games such as Dragon Quest or Kingdom Hearts on Switch by SquareEnix not because of some fictional change in the relationship between Nintendo and SE management but because SE is heavily using UE4 which is supported by Switch.