This is a pretty interesting question actually. I'm not as well versed on the history of the soft drink industry as I am of alcoholic beverages, so I'll approach it from the side I know best. And for the purposes of this, I'll define "fruit punch" as the red colored beverages made from concentrated syrups, most famously Hawaiian Punch.
The first root is pretty clearly the alcoholic punches of the 1600s - early 1900s. David Wondrich's Punch is an excellent book covering the history as well as recipes, and by combing through them one can trace the evolution of the medium. The first punches read more like what we might call "grogs" today. Lemons or Limes, sugar, and liquor: palm Arrack, brandy, or rum. Over time, these recipes evolved to include ingredients that showed off a host's taste and purchasing power. Oranges, raspberries, champagne, other french wines, pineapple (famously expensive in the colonial era), and many others.
By the end of the era, some of these fancy punches are sweet, fruity concoctions. Here's the Champagne Punch from Jerry Thomas's Bar-Tender's Guide (1862):
This is not wholly unlike the various "punches" which graced the serving boards of many american parties in the mid 20th century, once you substitute lemon lime soda for the champagne and lemons, sherbet for the orange, and Hawaiian Punch for the pineapple and fruit syrup. Wondrich makes a similar point when discussing the eventual fate of what he labels the capital-P "Punch". But by itself this doesn't answer the question, seeing as one of the ingredients in the concoction is itself a fruit punch.
I have seen claims that fruit punch was invented by Queen Victoria because she preferred her punch alcohol free, but I have yet to see a source. Additionally, these punch recipes would absolutely not work without their liquor component and would have to be totally re-balanced. It just doesn't add up to me and it reeks of clickbait factoid.
Next up is the shrub. Shrubs are basically a way of preserving excess fruits by soaking them in vinegar, straining, then sweetening and reducing to a syrup. This syrup was then diluted with water, or eventually soda water. Often rum or brandy would be added (the sheer volume of alcohol consumed in the 18th and 19th centuries boggles the mind and is well documented in W.J. Rorabaugh's The Alcoholic Republic.) I can make no claim as to whether it directly inspired fruit punch, but at the very least from a beverage construction standpoint, it's a clear ancestor: A fruit syrup that's diluted into a beverage. Shrubs were also used in cocktail making, fruit syrups were well and good for a Punch, where you could make what you needed for the application, but without the stabilizing agent of vinegar wouldn't last very long. As refrigeration technology began to be more available, the shrub largely faded from memory until resurrected by enterprising bartenders and history buffs.
Now we get to the actual thing itself. Soft drinks are not my area of expertise, and as such I'll stick to the basics here. As best I can find by canvassing the origins of the fruit punch brands, Hawaiian Punch is the first to market. Kool Aid is invented in the 20s, but was not available in a mixed "fruit punch" flavor originally. Hi-C and Minute Maid are both in the 40s. But Hawaiian Punch is first made in the 1934 by the Pacific citrus products company in Fullerton, CA. Not originally a prepared beverage, but a syrup intended to be poured over ice cream. Consumers realized it could be mixed with water to make a sweet and popular beverage, and was then sold as such. The original ingredients were pineapple, orange, passionfruit, guava, and papaya, which the brand website claims were imported from Hawaii, thus the name.
Now, here's the curveball: fassionola syrup. And because it involves the most secretive and difficult to research aspect of Alcohol history, tiki drink making, unfortunately it doesn't have a definite conclusion. Fassionola syrup is a red fruit syrup which was reportedly primarily flavored with Passionfruit and other fruit juices. The early tiki bars notoriously wrote their recipes in code to prevent bartenders from stealing them, and as such, some things remain unclear to this day (see the work of tiki bartender and researcher Jeff "Beachbum" Berry for more on this, he is the one who did the legwork of tracking down old bartenders and managers to piece many recipes together).
Donn Beach (born Ernest Raymond Gantt), founder of Don the Beachcomber's, is claimed to have invented it, but Donn was somewhat famously a liar and showman, and it's hard to take stories about him at his word. Now, the first Don the Beachcombers opened in 1933 in Hollywood, CA. Drinks that were served during the 30s there were later said to use fassionola syrup as an ingredient (remember, the secrecy of the ingredients means I have to use words like "said" and "claimed", and it's very possible that these recipes were later adjusted to include this ingredient). The syrup was commercially produced for a time by a California company, and those who have tasted those older syrups (including the previously mentioned Jeff Berry) claim it tasted identical to Hawaiian punch concentrate.
Now, note those dates and locations, and compare to hawaiian punch and the Pacific citrus products company. Two red tropical fruit syrups were produced independently by two separate groups geographically close to each other? It's certainly possible, but it would be a pretty big coincidence if so. I'm not going to speculate, but it certainly muddies up the waters. I have seen claims that both were inspired by Passionfruit, Orange, Guava juice mixtures from Hawaii, but I myself have been unable to verify if that was indeed available then.