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it's made using a process called the "Champagne method" or "méthode champenoise" which involves adding sugar and yeast to the wine and then bottling it before fermentation is complete. the yeast consumes the sugar and produces alcohol and carbon dioxide, which gets dissolved in the wine and can't escape because the bottle is sealed. this increases the pressure inside the bottle, making the wine bubbly. when you open the bottle, the pressure is released, and the carbon dioxide bubbles rise to the surface!

those bubbles appear to come from a specific spot because that is where the pressure inside the bottle is the greatest, which is typically the spot on the bottom of the bottle. hope that helps!

Thank you so much for such an informative answer, it’s truly fascinating and couldn’t for the life of me formulate any idea on how it works! So essentially, those bubbles are yeast farts?

yeast farty party would've been a much more concise answer

The phrase you're looking for is "bubble nucleation." Bubbles don't really just form spontaneously. They need something to start the process, after which they grow larger until they are big enough to float upward. If there's a single place where they seem to form more consistently, it's because there is something different about that spot. Most likely, there's a small irregularity in the glass there... probably a small bump. That irregularity makes it easy for bubbles to form.

In a similar vein, when you're doing lab chemistry, there are many chemicals you want to boil in the most controlled way possible because they're dangerous if they get out of control. One way to do that is to deliberately introduce something into the beaker that will allow the bubbles to form smaller, faster, and more easily. "Boiling chips" are added to accompany this. They are small pieces of an inert substance that have lots of edges and points (much like the glass irregularities in the champagne bottle or cup that you observed.)

Here is a YouTube video that shows some boiling chips in action.

Bubbles form easier at "nucleation sites", which are often tiny imperfections in the glass. Part of the reason why cola light explodes when mentos is added, is that the candy surface is especially accommodating to nucleation.