Winemaking is a tradition that's thousands of years old. Although grape wine is definitely the most popular even today, it's not the only way to make wine. As you've noted, you can make wine out of all sorts of stuff, ranging from peaches to dandelions.

So why grapes and not dandelions? It's pretty simple: grapes have a ton of sugar, and when yeast eat the sugar they end up producing alcohol as a byproduct. As the Joy of Home Winemaking notes:

The grape is the only fruit that is sweet enough to provide enough sugar and acid on it own (and in the olden days, its own yeast) to make that much alcohol

If you try making peach wine the same way you would make grape wine, you'll technically end up with an alcoholic beverage, but the actual amount of alcohol will be so low that it will spoil quickly. If you look up recipes for fruit wines, you'll notice that nearly all of them involve adding large amounts of sugar in order to give the yeast enough food to produce a sufficient amount of alcohol. Many recipes also call for adding some acidity to increase the durability of the wine.

This was important because there wasn't really a great way to preserve wine. Wine bottles started becoming much more feasible in the 17th century with the invention of the coal furnace which allowed the creation of thicker and more durable wine bottles. Until then wine was stored in barrels which were not air-tight and therefore prone to spoilage.

Honey wine (aka Mead) was the big alternative to non-grape wine as it's also sweet enough to naturally produce enough alcohol without any additional sugar. Honey is considerably more difficult to harvest than grapes, so production was never as widespread.

While grape wine and honey wine were the big two varieties of wine, people did experiment with adding other fruits and edibles as adjuncts. The Greek physician Pedanius Dioscorides details a lot of variations on wine in his work De materia medica. These range all the way from Oinos Roites (Pomegranate Wine) to truly unusual stuff like Strobilites (Pinecone Wine):

Strobilites is made from new bruised pinecones steeped in must [grape pulp]. It has the same effects as resinatum [5-43]. Pinecones steeped in must and boiled are good taken copiously as a drink for pulmonary tuberculosis

As I noted before, trying to ferment pinecones by themselves into wine is not going to work every well, the grape pulp is doing most of the work in terms of converting sugar into alcohol.

Eventually around 1800 a French chemist named Jean-Antoine Chaptal invented a process called "chaptalization" (named after himself) where sugar is added to the wine must before or during fermentation. This allows fruits (as well as grapes that did not ripen well) to still produce viable wine. Not coincidentally, he was also part of a committee tasked with cultivating sugar beets that which allow for a substantial amount of sugar to be extracted. From The Sugar Beet: Including a History of the Beet Sugar Industry in Europe:

In Archard's report he consider that there could be no doubt that it was possible to extract the sugar for the beet...a second committee of chemists was formed, composed of Chaptal, Crespel, Delesse, Barruel, and Isnard. These gentlemen's ideas combined led to many improvements on the processes that had heretofore existed.

Thanks to the work of Chaptal and other chemists, sugar had finally become cheap to produce in such quantities that fruit wines were actually viable. And people did experiment! A good example of this is the Victorian guide for homemakers Mrs Beeton's Book of Household Management which was published in 1861. In the book, Beeton details recipes for several non-grape wines. Notably, she makes use of chaptalization which differentiates her recipes from those of Dioscorides: her wines truly don't have grapes at all. For example, her recipe for Dandelion Wine:

Ingredients.—4 quarts of dandelion flowers, 4 quarts of boiling water, 3 lbs. of loaf sugar, 1 inch whole ginger, 1 lemon, the thinly-pared rind of 1 orange, 1 tablespoonful of brewer's yeast or ¼ of an oz. of compressed yeast moistened with water.

Method.—Put the petals of the flowers into a bowl, pour over them the boiling water, let the bowl remain covered for 3 days, meanwhile stirring it well and frequently. Strain the liquid into a preserving pan, add the rinds of the orange and lemon, both of which should be pared off in thin fine strips, the sugar, ginger, and the lemon previously stripped of its white pith, and thinly sliced. Boil gently for about ½ an hour, and when cool add the yeast spread on a piece of toast. Allow it to stand for 2 days, then turn it into a cask, keep it well bunged down for 8 or 9 weeks, and bottle the wine for use.

So fruit wines have been feasible to produce for well over a century. Despite that, the vast majority of wine produces and consumed today is still grape wine. And while I could speculate about why grape wine is so much more popular, it'd just be my personal opinion, so I'll leave it as an exercise to the reader to come up with why they think grape wines are so much more popular than other fruit wines.

I can address some, but not all, of this question. I'll add my perspective to what u/cjt09 posted a bit ago.

On the face of things, I'm a history teacher with a specialization in the Protestant Reformation. But I have a dual appointment, and I also teach in the Biology department (long story). Additionally, I brew beer, wine, and cider on the side. So, parts of this question are right up my alley. For what I can't answer, I'm afraid you'll have either narrow the scope of your question or wait for someone who knows more about post-Prohibition alcohol consumption patterns to come along.

A great deal has been written in other sources about both biology and chemistry of fermenting fruit, usually including terms like "fermentable sugars," "tannin content," and "pectic enzymes," claiming that grapes are inherently superior to other fruits for fermentation. I don't buy that. For one thing, wine drinkers prefer the taste produced by the specific biochemistry of grapes because it's the taste produced by grapes, and people tend to prefer the things that they are used to. Wine is an acquired taste, and people can and do just as easily acquire tastes for other beverages with very different flavor profiles. For another thing, the grapes that are grown today for winemaking have been bred over thousands of years to achieve particular characteristics. Had people put a similar amount of effort over the centuries into developing new varieties of, say, cherries that had a similar tannin content (for example) to cabernet sauvignon grapes, it is very likely they would have been successful. So the inherent "superiority" of the grape is not the issue here.

People around the world make alcohol with whatever fermentable materials they have. In Japan they make sake from rice; in Scotland they make whisky from barley; in Southern Africa they make wine from palm trees; in Central Asia they make kumis from horse milk; the medieval Norse in Scandinavia made mead from honey; in the Amazon they make chicha from manioc; in Hawaii they make okolehao out of an indigenous plant called "ti;" in Mexico they make mescal from agave. In all cases, they make X from Y because it grows there. Grapes grow well in a Mediterranean climate, which happens to cover much of Southern Europe, Southeast Asia, and North Africa. Not many other fruits were widely grown there until more recent times. So, the short answer to your question is that European people fermented what they had (grapes), and acquired a taste for the product (wine), so then when they spread themselves, along with their culture, tastes, etc. to other places wine became popular there as well.

I don't accept your premise, though, that grape wine historically was much more popular than alcohol made from other fruits throughout the western world. In today's era of mass production and globalization that may well be true (the United States consumed 805 million gallons of wine in 2018, according to this website and 59 million litres, or 15 million gallons, of cider in 2012, according to this one), but it was much less the case historically.

Wine was not the most popular fruit-based alcoholic beverage in much of Northern Europe historically, nor is it in some areas there still today (to say nothing of the rest of the world, but your question mentioned Western culture). Cider (made from apples) and related drinks made from pears and quinces were more popular in the cooler, wetter climate there. Grapes grow well in Italy, Spain, and southern France, but places like Normandy, Britain, and much of Germany are better suited for apples. The same was eventually found to be true in New England and the Midwestern United States. Grapes don't grow well there, and wine was expensive to import. Cider was the preferred fruit-based drink of both the colonists and the citizens after independence, up until the temperance movement and prohibition upended Americans' drinking habits.

The point of fermenting fruit is to make alcohol (preservation is a side effect). Grapes contain more sugar than apples, and so produce a beverage that is more strongly alcoholic (10-15%) than typical apples or pears (~5%). In that sense, grapes are inherently superior. But the same cold winters that make places like New England bad for grapes and good for apples also allow cider makers to freeze concentrate their product. Simply leave a bucket of cider outside on a cold night, remove the ice that forms, and repeat until the cider doesn't freeze anymore. The water freezes and the alcohol remains, eventually leaving the cider maker with a much stronger product called "applejack" that can exceed 20% alcohol content.

Additionally, in cider growing regions people put just as much effort into developing cider apples as people elsewhere put into developing wine grapes. They aren't as widely planted or consumed today, but apples like golden russet, brown snout, and foxwhelp have poor flavor and texture for eating but excellent characteristics for cider. Perry is made from specific pear cultivars like merrylegs, mumblehead, and stinking bishop (yes, you read those right).

So the supremacy of grape wine is recent, at least in regions that do not grow grapes. As I mentioned briefly above, the dramatic decline of cider production in the United States, at least, took place in the context of the temperance movement. The recent popularity of wine is related to globalization; I don't know where you live, but if you walk into any liquor store I can almost guarantee you'll find wines from France, California, Italy, Australia, Spain, Argentina, and South Africa, all starting at a few dollars a bottle. This has made grape wine much more accessible for a great many people who centuries earlier would have consumed locally made beer or cider.

Edit: typo