PuTTY, probably not. But if you're using Windows, the built-in Terminal supposedly supports panes: https://learn.microsoft.com/en-us/windows/terminal/panes

Combine that with the built-in ssh client, that might be good enough for you. Personally, I have to run Windows at work but I'm allowed a local linux VM, so I run that headless, ssh into it, and run tmux for pane/window support.

For sessions on the same target host you could use tmux.

You can ssh from emacs.