The fog is exactly the same as any fog: a collection of tiny particles of liquid water that have been condensed out of the air by a cold temperature, but are too tiny to fall down like rain. They are kept suspended by being constantly bombarded by air molecules. They look white because they reflect all wavelengths of light equally.
Your puzzlement apparently stems from the fact that you can’t see any fog inside the bottle until you open it, yet it is equally cold at both times. What is there about opening the bottle that makes the fog form?
The space above the beer in the unopened bottle is filled with a mixture of compressed carbon dioxide, air, and water vapor, all gases. The water molecules in the vapor are content to stay that way, far apart from one another as an invisible gas, rather than clumping together as particles of fog.
They do so because they got there in the first place simply by leaping out of the beer’s surface, and at the temperature of the beer, only a certain number of them, and no more, will have had enough energy to leap into the void.(Techspeak: The water vapor is in equilibrium with the liquid at that temperature.) The water molecules remain that way until you come along and upset everything by removing the cap and releasing the pressure.
When the pressure is released, the compressed gases are suddenly able to expand, and when gases expand, they lose some of their energy and are cooled. The gases are now cold enough to condense out some of the water, and that’s the fog that you see.
If the bottle is put down in front of the customer and not poured immediately, you may see some of the fog actually rising above the mouth of the bottle and spilling over onto the bar. Dissolved carbon dioxide gas is now leaving the beer and expanding as it hits the warmer air at the top of the bottle.
As it expands, it lifts some of the fog. Then, since carbon dioxide is heavier than air, it actually spills over like an invisible waterfall, carrying some of the fog down with it along the sides of the bottle.
No offense, but if you worked in a higher-class establishment you’d notice exactly the same fog effect upon opening bottles of champagne, and for exactly the same reasons.