What is this suspicious stain under a VFD's glass?

It's getter, it should absorb any residual gases in the display. Vacuum tubes have getters too (the black part on top).

If that spot turns white (though in some cases I have seen it become transparent/invisible or just fall off) it means that the glass is broken and the VFD or tube is full of air and is no longer functional.

It is called a getter. It's design to be a reactive deposit to absorb the remaining air in the vacuum. The ones in the linked photo are probably fine, the Wikipedia article has photos of examples where the vacuum has been lost (they become completely whitened).

It is as designed. No worries! It's called a getter. It'll get the remaining O2 that was left in there.