I don't think it's so much about whether Nintendo has the right as opposed to whether one should do business with them based on how overly defensive they are. There are other gaming platforms in town, and none of them (not even Sony) are as needlessly defensive of their IP as much as Nintendo, so it isn't obvious that Nintendo is just doing what it takes. They go above and beyond.

Also, whether a crime is being committed at the same time as displaying a trademark shouldn't matter unless the trademark is actually being violated. Since trademark today is effectively being enforced by multinational corporations scratching each others backs, we may never actually know whether Nintendo's lawyers could convince a judge or jury.

Good question. I don't actually know, although I'm not sure how Japanese law would extend to a multinational social media platform that happens to be headquartered in the United States.