Is the resulting file supposed to executed under Cygwin ?
No.
If you build Make natively for Windows, this results in native Windows version.
Cygwin is actually quite a different story. Cygwin is not really Windows. Perhaps you heard the word "personalities". Windows NT in the beginning was supposed to have multiple "personalities" (read: APIs) to be compatible (sometimes even on binary level) with different OSes. These personalities included WinAPI (actually old 9x API), OS/2 (binary-compatible) and UNIX (only source-compatible because still relied on PE format, not ELF).
With time Microsoft decided to phase our everything except WinAPI. Cygwin is actually modern, opensource version of UNIX personality.
When you build a program for Cygwin, you use Cygwin compilers, not Visual Studio or MinGW. Cygwin appears to be a different platform with a different runtime. You cannot compile for Cygwin using Visual Studio (well, in theory you can, given you supply a set of proper includes and libraries), but i don't know about practical attempts to do that.