The kill ring is part of Emacs. The operating-system clipboard can connect it to other applications when the display session and configuration support that connection. Under normal graphical clipboard integration, M-w copies to both, and C-y can insert text copied more recently outside Emacs. That is why checking the pasted result matters even after checking the selection.
Test the boundary with a short, non-confidential sentence in each direction: copy from Emacs and paste into a disposable external document, then copy a different sentence externally and paste into an empty Emacs buffer. Verify the actual text rather than assuming that success in one direction proves both. Native Windows Emacs, WSLg, terminal Emacs and remote sessions need not share identical clipboard behaviour. This guide does not install a bridge or change your clipboard configuration.
If copying between Emacs buffers works but the external test fails, investigate that display/clipboard boundary rather than changing speech settings. Do not paste test commands into a live terminal. For the general integration and its options, see Using the clipboard and Cut and paste with other window applications in the GNU Emacs Manual; a primary selection is not the same as the clipboard.