Carbon dioxide production is a byproduct of the yeast that ferments sugar to alcohol. Soft drinks are carbonated by adding CO2 under pressure, but for an alcoholic drink it's enough that it's fermented in a closed container. The in-bottle fermentation that gives champagne its carbonation is what set it apart from other wines, historically (although today 'sparkling wine' is the general term and 'champagne' only refers to those that come from that region of France. Beers could continue to ferment in their barrels, since they didn't filter out the yeast as much then.

Spring water is sometimes carbonated as well, in those cases from dissolving carbonate from rocks underground where there's high pressure, the same pressure that eventually pushes the water up to the surface, but artificially carbonated water and soft drinks are 19th century inventions.

Winemaking is very simple (it's essentially just old grape juice) and - in its essentials - has largely stayed the same, apart from improvements in quality. But those were very large improvements (and it's not one continuous tradition of winemaking around the world). Wine tasted very bad in the old days, one of the main reasons being that they couldn't store it well, and so it got vinegary. So they often added things to make it more palatable, etc. So if you go back to biblical times, their wine is very very different from what we have now. If you go back 500 years, the difference is not so great though.

Beer is more complicated, as it's made from grain that must be malted before it can be fermented, and they're normally flavored with hops if not other things as well. There are more different kinds of beers than wines, although on the other hand, in modern times beer production has achieved levels of consistency that doesn't exist for wines, and did not exist in the past. You can rest assured that next year's Carlsberg will taste indistinguishable from this year's. That's not true of wine, nor of beer that's produced with more traditional methods (in which case it's really the batch and not year that matters).

There are beers on the market that could represent many places and eras, but if I were to mention one, unflavored lambics are perhaps the most 'primitive' tasting beers. These are beers made without any specific yeasts, just whichever wild yeasts and bacteria happen to get into the containers. It's a bit of an aquired taste, to say the least (and there's a reason most of them are flavored). They still have hops added though, making it a step or two up from the most primitive possible kind of beer. But it'll take you back to the middle ages at least.