Maybe I'm being naive, but I'd think this isn't really about Nintendo paying for promotion but more about Japan looking for universally loved characters to make a statement to the world. It reminds me of the James Bond skit with the queen in the London 2012 opening ceremony.
I don't think you're being naive at all, and that that is indeed what happened. It's obvious from what characters were involved in the ceremony that they were chosen as ones that best represents Japan's imagine of a modern and futuristic world. They even talked about showing off futuristic technology as a theme for this ceremony in one report I read. What better way then to get those big stars people around the world know and love. That's why you saw doraemon, and pac man, and of course Mario. They almost certainly went to Nintendo, asked to use him and they said absolutely sure, cause why wouldn't they? This is a huge event for Japan, a big moment of pride for the nation that's had to deal with some frustrations the past few years, and everyone wants to be a part of this great moment. I can guarrentee you, no one at any anime message board right now is going, "oh I can't believe the doraemon people are wasting their funds doing the rio ceremony."
And you know what, it's even more silly as even though I think it was as said above, they were asked to participate and they did for the sake of Tokyo 2020 alone, Mario has been a part of the Olympics for almost ten years now with all the Olympic themed games. It's a part of the Mario universe now and has been for a while. It's not like they just jumped into this now. They have invested into the games for a long time, and it's obviously doing some good as they continue to do so.
Either way it's silly. This was a great moment for everyone involved. And it did what it was supposed to do, show off The Japan the world knows and is looking forward to.