In any case, I welcome a posix interface for windows, it would be a cool tool to have, especially in the case of cross-platform utilities.
Aside from that, Cygwin tries to hard to be a complete Unix environment on Windows, whereas midipix just gives you enough to use interfaces that were standardized in POSIX as a reasonable, uniform API for all operating systems to provide. Some functions go beyond that, but you don't have to use them. And even some things that are mandatory in POSIX are optional in midipix; as I understand it, you can choose at build time whether you want the overhead of being able to support tty devices (and the associated semantics like job control, signals from the controlling tty, etc.).
Cygwin works fine with multiple installations these days [https://cygwin.com/faq/faq.html#faq.using.multiple-copies]
The thing that wont work is if you try to mix and match a dll from one installation with binaries from another, but I think you can agree that that situation is fair. You just need your paths setup correctly.
Cygwin and MingW don't evoke any existing meaning, so I think they're somewhat more memorable.