New terminal window (duplicate session) from Putty command line?
In Putty Settings > Window > Behavior, you can check one of the boxes to open the system menu on a certain keypress (I personally use ALT-Space).
With this setting in place, you can hit ALT-Space, then type the d key to Duplicate Session. This will allow you to open a new putty window without needing to touch the mouse.
It's not a command line tool, but I find it extremely useful.
There is no straight forward way to issue a command on a Linux host through ssh that will instruct the windows host where the ssh connection originated to spawn a new putty instance.
The remote host knows about putty only that it is a terminal capable of running a certain shell. It's not supposed to know how to spawn a new terminal on the local client.
Conceivably, it is possible to throw together a script of sorts (or look for a windows netcat clone) that will listen for a "ping" and spawn a new Putty. The second part of this "system" would contact the windows script via TCP from the remote host.
If you don't want to touch the GUI, the only thing I can think of is using a terminal multiplexer like
tmux. This wouldn't give you another GUI window, but you could have multiple windows/panes inside of either.