De manual :
3.1.3.7 Full screen mode
If you find the title bar on a maximised window to be ugly or distracting, you can select Full Screen mode to maximise PuTTY ‘even more’. When you select this, PuTTY will expand to fill the whole screen and its borders, title bar and scrollbar will disappear. (You can configure the scrollbar not to disappear in full-screen mode if you want to keep it; see section 4.7.3.)
When you are in full-screen mode, you can still access the system menu if you click the left mouse button in the extreme top left corner of the screen.
Parece que você encontrou outra forma de Ctrl + clique com o botão direito do mouse.
Você pode habilitar a tecla de atalho usando Alt + Enter como mostrado na captura de tela anexada:
Mais uma vez no manual :
4.9.7 ‘Full screen on Alt-Enter’
If this option is enabled, then pressing Alt-Enter will cause the PuTTY window to become full-screen. Pressing Alt-Enter again will restore the previous window size.
The full-screen feature is also available from the System menu, even when it is configured not to be available on the Alt-Enter key. See section 3.1.3.7.