I worked in refrigeration for most of my life and I have a theory how this could indeed work. When air is compressed the heat that is in that air is pushed into a smaller area, this raises the temperature. (You can easily see this on any air compressor, the discharge line is hot). if you remove the heat, then release the air the expansion causes a cooling effect, ( the expanding air absorbs heat).
In this case the compression is minimal, but it does exist. The heat of compression would be right at the neck of the bottle where it is fastened to the thing that holds the bottles in place. The heat would thus be transferred from the bottle to it's holder. The air is not only blowing into the bottles, but also on it's holder.
So if the holder was even a little bit warmer than the outside air that is blowing against it, the heat would be transferred from the holder to that air that was blowing against it. Once the heat of compression was removed, and the air left the bottle neck it would expand and absorb heat in the process causing a temperature drop. Maybe only a few degrees, but even that can feel good.