From personal experience (this will probably be deleted, as I don't have the required books in front of me), wine can be made out of anything with sufficient juice or sugars. I'm going to speak as a NASA lvl2 sommelier and an amateur music historian focusing on early (1950s-1960s) American country.

Any fruits can be juiced and the juice fermented into wine. In addition to that, any product with enough sugars or starches can be fermented into wine. In east Asian societies, the most notable of this is sake, a rice wine. There's also soju, a South Korean drink made of rice, wheat, barley, potato, or sweet potato. Likewise, Britain and France have dandelion wines.

In regards to your question about 1950s and on classic rock or country music, there's two different types we can talk about. The first is simple, and that is "Boone's Farm" style wines. These are kind of like wine coolers, in that it's a grain neutral spirit or alcoholic malt beverage added to juice or juice flavors. They are cheaply made and cheap to buy, the most popular flavor of which is strawberry (due to the sweetness of it). The second would be homemade fruit wines, with the sugar and alcohol content varying based on the wine makers' taste, knowledge, and expertise. When a fruit crop wouldn't sell, due to ugliness, quality, ripeness levels, etc, it's very easy to juice said fruits and turn them into wine to at least get some money from it.

To put the second category into perspective, let's say you had a peach orchard. If 3/4 of the peaches were good and sold at market quickly, you are still left with 1/4 of them being misshapen, too ripe, etc, and you don't want them to go to waste. That's wasted money and income. So now you can juice them, and have some options. You can sell the peach juice in jugs or can sell it to frozen juice companies (if you had a contract with them). Most rural people in this situation would go an alternate route and take their chances on some alternative income - they'd ferment the fresh juice with some yeast and make a homemade wine. Even if they couldn't sell at market or create a "winery" of sorts, there was still alcohol for them to drink.

So now we've circled back to country and early rock music. Most 1950s and 60s country (and by extension, rock) musicians didn't have a rich upbringing. Their families weren't typically wealthy, and they were not usually living in cities with access to classical or jazz music. Few had any formal music training outside of church services. Very few popular songs from the 50s and 60s dealt with fantastical lands, and country in particular was more about what somebody saw and experienced. Take Dolly Parton's "Coat of Many Colors" or "Coal Miner's Daughter" as examples - a poor girl growing up, and the struggles she experienced. Now, if you as a teenager in that situation were able to get some money scraped together, would you buy a vintage bottle of Napoleon cognac or the cheapest alcohol you could find? And what would be more resonant with the populace? "My Butler Brought Me Louis XVI Brandy When I Wanted Napoleon" or "I Found A Dollar And Bought Thunderbird Wine"?

Finally, these wines are still common in the United States, but not likely to be found in liquor stores. There's still plenty of farmers markets and small towns with different wines you can find, though few would be available nationwide. My solution is to get out and explore orchards in your area and see what is available.