Does cygwin work line wine, providing a compatibility layer inside a foreign os?
Does Cygwin work like wine? No.
Does it provide a compatibility layer inside a foreign OS? Yes.
Wine can run Windows executables on Linux, but Cygwin cannot run Linux executables on Windows. Instead, Linux programs have to be compiled specifically for Cygwin, whereby the aim of the Cygwin project is to make that as straightforward as possible, i.e. it's aiming for source compatibility rather than binary compatibility.
However, Winelib, which is part of the Wine project, essentially is Cygwin in reverse: it provides a source compatibility layer that allows Windows programs to be compiled into Linux executables.