The DMCA is a broad set of laws that includes, a process for copyright owners to ask publishers to take down data. This is defined in sect. 512(c) of the copyright act, and it comes with some requirements from the claimant side of things (here: Nintendo), and some liability on the publisher side of things (here: Valve). It also includes rights for the entity accused (here: the Stichting Dolphin Emulator) to counter claim, allowing the publisher to reinstate the content until the claimant sues.
In this case, none of this process was followed. To the best of my understanding, this is what happened:
1. Valve legal contacted Nintendo of America to ask "hey, what do you think about Dolphin?"
2. Nintendo replied to Valve "we think it's bad and also that it violates the DMCA anti-circumvention provisions" (note: nothing about violating copyright itself). Also "please take it down".
3. Valve legal takes it down and forwards NoA's reply to the Dolphin Foundation contact address.