Pipe putty context to client clipboard?
There is a putty patch doing exactly what you want.
http://ericmason.net/2010/04/putty-ssh-windows-clipboard-integration/
And this patch has been integrated into kitty.
If you want to copy all the putty output to clipboard, there is a "Copy All To Clipboard" option available in context menu (right-click) from title bar of the putty window.
2020 Answer. As mentioned in previous answers it is possible. Idk how about official Putty because i use KiTTY (enhanced version of Putty) instead.
But all you need is to set Windows clipboard
in Terminal > Remote-controlled printing
and then use following code as a bash script or a function or whatever you like and then pipe into it.
echo -ne '\e''[5i'
cat -
echo -ne '\e''[4i'
Usage example: cat <whatever> | puttyclipboard.sh
PS: The content (and the original idea) of the dead website ericmason.net mentioned in previous answer can be found at web.archive.org