So for a developer to release a game on the Game Boy without Nintendo knowing, they would have to commit copyright infringement.
That'd be trademark infringement, not copyright infringement.
They used this same tactic on the Switch. They claim the prod keys, which are needed for Switch emulators, are copyrighted.
That's not quite the same thing, and still isn't because the keys are copyrighted. There's Digital Rights Management software running on the Switch, and part of what it does is decrypt encrypted parts of games with the keys. Originally, Nintendo managed to keep the keys secret, but eventually people managed to extract them. The next line of defence is that under the DMCA (or equivalent law in countries with a trade deal with the US), it's illegal to attempt to circumvent DRM, and as the keys are capable of doing that, they themselves might count as a DRM circumvention device, which would be illegal to own or share. It's a legal grey area whether or not they'd really count - lots of companies claim that it's illegal to have these so-called illegal numbers, but Wikipedia are confident enough that that's not what the law really says that their Illegal Number page lists a bunch of them.
This gets even more complicated when it's specifically about emulators, as the DMCA (or equivalent) have a specific carve-out for interoperability, saying you're allowed to ignore parts of the DMCA if it's specifically for the goal of making computer software work with computer hardware it wasn't originally intended to. For the relevant parts of the DMCA that aren't related to DRM, there's case law confirming that it's okay. However, no emulator developers have ever actually been sued for making an emulator for a system with any DRM (e.g. the thing with Switch emulators several months ago was settled out of court, and the threat was to sue them for things like illegally sharing games between developers, when they could have each bought their own copy, so weren't protected by the carve-out). That means that this is a grey area, too.
If Nintendo wanted to shut down an emulator based on its use of their keys, they'd not only have to set a precedent that the keys really did count as a DRM circumvention device, but also that the interoperability carve-out didn't apply to DRM circumvention devices. It would be a big, expensive case, and as there are well-funded organisations that rely on the precedent not being set against them in both directions, both sides would get interested third parties funding their legal fees. No one wants that, so Nintendo stick to claiming emulators are illegal on their website, not in court documents, and only go after emulator developers who've provably done a second illegal thing they can be punished for.
