Putty (or Kitty or MobaTerm) is still a thing, but there are other options. The Linux subsystem will give you a *nix shell and ssh that works exactly like you expect it to. There is also msys which will give you an ssh client you can run directly from cmd.exe or power shell. Microsoft have been working on a proper native port of OpenSSH, but while it apparently works pretty well, it's still not feature complete or ready to be shipped as a standard part of Windows (hopefully early next year).

I use putty. Haven't found a compelling reason to use anything else yet.