So the question is: Is it fake advertising?
but I just do not understand why it is not for sale here
or mentioned in any way.
It's certainly no lie that you can use an Arduino to display data on a TV or a VGA monitor. I am a bit confused by your statement about a VGA output being used to display data on a TV though. I'm not very knowledgeable on these matters, but my understanding is that VGA was only used by computer monitors, not by TVs.