IMHO you have managed to get into a situation where Windows has been asked to run two different cygwin1.dlls at the same time and Cygwin has detected this. (I recommend using Process Hacker 2 for helping to track down which processes have handles to cygwin1.dll).

This is unfortunately just something you need to manage for yourself.

Cygwin got itself into trouble like this because it never had a good enough package management story (because Windows can't overwrite a currently loaded dll, they didn't attempt a good system package manager, as best I understand it) so developers would end up bundling cygwin1.dll alongside their own executables instead, so there would end up being lots of cygwin1.dlls on a users machine.

The same thing could happen on MSYS2 if you had two MSYS2 environments, however, to get them both in-sync again, you'd just have to do "update-core" in each and that would be that.