Disclaimer: I'm not officially involved with Dolphin anymore. I was the treasurer for the foundation backing the project for a while (technically still am for a month), but I've stepped down from the project a month or so ago. So, still plenty of context, but not much at stake for me.
The error that many have done in their reporting is to say this was a "DMCA takedown notice" or "DMCA notice" or (ugh) "DMCA". This was none of these things.
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.