If a copyright holder discovers infringing content on our site then it's entirely in their right to file a DMCA takedown notice. We would comply, and take the content down. We'll of course notify the developer of their legal options. A counter claim is possible, but it would be responsibility of the uploader.
Nintendo is responsible for protecting their intellectual property and if those fan games were left up it could set a unfavorable precedent for Nintendo if they need to protect their properties in the future. Fan games are a gray area though, since one could argue that they are in fair use. (Although I have not seen the games that were requested to be removed, it's possible they were using stolen assets.) I have a feeling they mentioned the advertising revenue in the takedown notice to add a stronger legal case if it needed to be taken to court. Online service providers loose safe harbor protection if they're getting revenue from the infringing content.
Will something like this happen to itch.io? I don't know, it's not for us to decide! There are plenty of cases of companies letting fan works exist, and other cases where they fight it. We're not responsible for upholding their copyrights, that's why a system like DMCA was put into place.