Oops, one more part of the cygwin patch (SF patch #102409 by jlt63:

Cygwin Python DLL and Shared Extension Patch).  Add module.dll as a
valid extension.

jlt63 writes: Note that his change essentially backs out the fix for
bug #115973. Should ".pyd" be retained instead for posterity?

Python/dynload_shlib.c

@@ -29,8 +29,8 @@
 
 const struct filedescr _PyImport_DynLoadFiletab[] = {
 #ifdef __CYGWIN__
-	{".pyd", "rb", C_EXTENSION},
 	{".dll", "rb", C_EXTENSION},
+	{"module.dll", "rb", C_EXTENSION},
 #else
 	{".so", "rb", C_EXTENSION},
 	{"module.so", "rb", C_EXTENSION},