Cygwin is a POSIX implementation, not a Linux implementation. Linux is mostly POSIX compatible and so Cygwin is source-compatible, but not binary-compatible, with many programs that work on Linux.

What the other guy said. Cygwin is a set of libraries to compile and run GNU programs on Windows. It does not run existing Linux binaries and definitely does not translate Linux syscalls into Windows (that would be WSL v1). If recompiling is OK, I suggest starting with one of the BSDs, at least they have a fork() that works.