PuTTY and mintty

Both PuTTY and mintty are not console applications, they are independent terminals providing connections to remote hosts or your local cygwin/msys installation.

Being ChildGui applications they do painting on their window canvas, instead of using standard input/output as console applications do.

That means that:

• neither of ConEmu terminal customization options may be used, these applications have their own configuration and canvas;
• neither (almost) hotkey configured in ConEmu may be used.

To avoid limitations it would be better to use real console applications (bash, zsh, ssh, etc.) from your favorite POSIX distro (cygwin, msys, wsl).

ConEmu creates by default Tasks for all found installations, just use them.