That is simply untrue. Wrong. An incorrect inference.
Many things get people drunk - alcohol, ether, nitrous oxide, even ordinary air if it’s breathed at an elevated pressure. Some of these agents are used for clinical anaesthesia - do you believe that they damage the brain?
The nitrogen in air is inert - it doesn’t react chemically with anything in the body, and your body and blood is full of it. And yet it is narcotic when breathed at elevated pressure, pushing extra nitrogen into your tissues. The mechanism is still unclear, but it appears that when extra nitrogen dissolves into the fatty sheaths around our nerves, it makes us drunk. This is called nitrogen narcosis and can affect SCUBA divers. It is also completely reversible when the pressure is reduced and the extra nitrogen is carefully allowed to escape from the body. No hangover, no brain damage, nothing. (You can get “the bends” if you let the nitrogen escape too fast, but that is a side issue. You can do the same trick breathing xenon mixed with air at atmospheric pressure, just as drunk, just as reversible, no problem with the bends.)
Whether alcohol acts by a similar mechanism is not clear, but there are some intriguing indicators. One is that intoxication can be “switched off” by increasing the atmospheric pressure - you can make rats too drunk to run around their maze, up the pressure and suddenly they’re sober. Take the pressure off and they get drunk again. This is called “pressure reversal”, and works for a number of narcotic agents besides alcohol. Does the increased pressure magically halt the “brain damage” caused by the alcohol?
Don’t get me wrong, alcohol can do a body a lot of harm. So can iron, vitamin A, even oxygen, if you have too much of them. But the mechanism of intoxication is NOT “brain damage”, and there is a level of alcohol consumption below which you won’t suffer toxic effects.
(Pressure reversal is briefly mentioned in: http://www.anes.upmc.edu/anesnews/volume/2003winter_spring/articles/focus.html - search for “pressure”. The rest of the article gives a fair bit of detail on the various theories of anaesthesia.)