In the case of fizzy drinks, carbon dioxide is dissolved artificially in the water at high pressure to give them this particular taste.
Closing the bottle (or the can) it is like loading a very small container with a lot of gas and squeezing it to allow more of it…like jumping on a luggage to allow more stuff in. No wonder that the gas is ready to come out!
When you open the bottle/can, the sudden change of pressure (the atmospheric pressure outside is lower than the one inside) favours the escape of gas, which come out in form of bubbles. A change of pressure, as we have shown at the stand, may create bubbles.
As you pointed out, however, there is something more than that;
Where do the bubbles form?
If you could look at the inside surface of the bottle/can with a very powerful microscope, you would see it is not flat. There are very small crevices where tiny bubbles are trapped. When you shake a can of coke, more of those bubbles are formed.
When the pressure change (i.e. the bottle/can is opened) those bubble expand (like a balloon taken on the top of a mountain) until gravity takes them away.
Therefore, the bubbles you see coming out “start” at those crevices (technically called “nucleation sites”) where pouches of gas of a few microns size are already trapped. Incidentally, this also happens when you boil water in a kettle: bubbles start to form at some specific places before covering all the hot surface! (you need a transparent kettle to observe this…)
This is why the experiment of the tapping works (see one of the movies on the Royal Society website): by tapping the outside of the can you destroy those small bubbles (by sending them back in the liquid) and, even after shaking, the fizzy drink does not come out. Try it at home!