Well, not quite. It was developed and published by Ubisoft, but you can't ignore that it is an IP collaboration, so it cannot be entirely third party. There are similar cases like Fire Emblem Warriors and Hyrule Warriors that were developed and published by Koei. M+R is published by Nintendo in Japan, FEW and HW are published by Nintendo in America and Europe.
You are looking at a combination of first and third party here and it's clear that none of the three games mentioned above can't go anywhere without Nintendo's consent.
If we evaluated a different, yet still similar case, what would you say whose game it is? The example I am talking about is Project X-Zone which is a collaboration between Namco, Sega and Capcom. Is it a Namco game? A Sega game? Or a Capcom game?
I don't think anyone is ignoring the fact that Nintendo licensed some of its IP to Ubisoft and Koei Tecmo for those games. But it doesn't change the fact that they were created, developed and published by third parties. Now, that being said, I'm arguing a real world definition. The "Switch Gamers Club" is your concept, and you are free to use a personal definition that is not in wide use if you so choose. Which is why I listed those games per your rules.