The problem is file system paths. The boot java process is started correctly from the domain.sh using JAVA path.
The same JAVA path is passed as a command line argument (jvm and default-jvm) to start other java processes from the running JVM.
Those attempts will fail with an invalid path error.
So, in case of cygwin, the values for jvm and default-jvm have to translated into the absolute paths.

Note, this can't be done to the JAVA path that starts the original JVM, it'll fail to start.