I want to start with an important distinction: the drunk foods/drinking foods that people were eating, versus the drinking foods that polemicists denounced people for eating. Unbridled alcohol consumption and excessive food consumption became strongly linked in Christian Europe under the banner of gluttony, a deadly sin, and we'll see that relationship play out through the centuries.

In early modern Europe and colonial America, when a lot of drinking was done at taverns that might double as boardinghouses or inns, it was often a question of luck of the draw. Were you in 16C Augsburg? Good luck, because outside of standard meal hours, your innkeeper wasn't supposed to serve anything but bread, cheese, and fruit. Inside meal hours, though, you might get some serious meat to go along with your beer or wine--butchers' guilds complain bitterly about tavern owners who were, against the rules, owning and slaughtering cows.

There was also the issue of quality. While English innkeepers had to be stopped from buying too much fish everyday, suggesting they couldn't keep food on hand long enough for it to spoil, apparently taverngoers in early America often ran into the opposite problem. In her diary from 1704-1705, Connecticut teacher Sarah Kemble Knight described her hostess preparing:

a twisted thing like a cable, but something whiter...[which she laid] "on the bord, tugg’d for life to bring it into a capacity to spread.

To disguise whatever it was, Knight noted, the tavernkeeper dumped Stretch Armstrong the mysterious white thing into a cabbage stew. (ETA: Sorry, thought that was a cultural reference y'all would pick up on.)

It wasn't just the food, either. Numerous accounts from 18th century America attest to the use of dirty bedsheets as tablecloths.

But people were definitely grasping the connection between eating along with food and not getting drunk as quickly. Reportedly, a 1793 drinking contest between expatriate British and French elites in Philadelphia ended quite poorly for the British, because Jean-Anthelme Brillat-Savarin made sure his countrymen ate enough to keep pace with the multiple varieties of wine and enormous bowl of rum punch--paid for by the Brits, of course.

So into and through the 19th century, there was somewhat less of a "specific foods" mentality and more of a "just make it A LOT OF FOOD." Scottish general James Edward Alexander (1803-1885), traveling through America, noted that tavern-goers ate every meal as if it would be their last:

Tables covered with meat, vegetables, preserved fruit, tea, coffee, and bread--both of maize and wheat--and soft hoe and waffel [sic] cakes

Patrons also ate like they were running out of time:

No ceremony was used; each man helped himself with his own knife and fork [from the serving dish] and reached across his neighbor...bones were picked with both hands, knives were drawn through the teeth with the edge to the lips...Beefsteaks, apple tart, and fish were seen on the same plate the one moment, and had disappeared the next!

When they finished, they retired to the next room for tobacco and "a stiff glass of sling from the bar-keeper."

One thing that seems to have been light on the menu: vegetables and fruit. Well, sort of. As Christine Sismondo archly puts it: "Fruit was usually drunk by the glass, in the form of peach brandy and apple cider."

Quantity over quality was the rule in the infamous hotel restaurants and saloons of the frontier and settling West, too. If you're interested, this page (scroll down) links to some menus from saloons that have been published in secondary literature.

One thing that was definitely not a drunk food in the later nineteenth century was dessert. The creation of ice cream shops and dessert/candy restaurants for women, specifically, was part of a larger attempt to keep fragile innocent women out of the men's drinking establishments. It's not an accident that these were foods more associated with children. (The history of how candy was made acceptable to men through marketing in the 20C is, by the way, fascinating.)

By 1900, the difference between "what drunk people ate" and "what polemicists claimed drunk people ate" was taking on a new and insidious dimension: xenophobia. There was rather a long history of American tavern food being very well sauced and spiced. And definitely salty, which Sismondo postulates was a conscious ruse to get people to drink more. Temperance advocates saw causality: "Condiments create a desire for narcotics," claimed one doctor. But everyone down to cookbook authors recognized the connection.

And in the burgeoning white nationalism of the late 19th century, good old fashioned ascetic impulses made a splash as a way to separate good white middle class Americans from the brown (especially Asian) newcomers. Spicy foods were denounced as too foreign, that horrible temptation of the Filipinos and Chinese. "Perverted appetites" was how Mrs. Norton's 1917 cookbook described it. Indeed, contemporary dietetic thought held that eating spicy food was not satisfying, because the spices camouflaged the good wholesome nutrition, and so people turned to alcohol to fill the void.

Of course, Prohibition activists themselves spent most of World War I harping that alcohol was food, or at least, that its use of grain was stripping food away from poor children and soldiers. (The medievalist in me needs to point out that in medieval Europe, yes, ale was an important source of calories for people who could afford to drink more than water). Danish-American journalist Jacob Riis certainly found the alcoholic version of wheat more palatable than the actual food served at the bars he haunted:

brick sandwiches, consisting of two pieces of bread with a brick between.

This was not, in case you are wondering, actually intended to be eaten.

The Prohibition era gave rise to its own brand of moralizing racist/alcohol/food polemicists. They ranted vitriolically about how black and white patrons mixed at underground speakeasies in Harlem, drinking and eating and maybe having sex. Others took a more pragmatic approach, publishing "glossaries" to help (mostly) white people understand the sex and food slang of the new scene:

Chitterlings is a tripe-like food, made from the lining of a pig’s stomach; Snouts are pickled pig’s snouts, a popular delicacy (New York Sunday News, 1929

The repeal of Prohibition brought with it various attempts to at least restrict consumption and sale of alcohol--still under the guise of moralizing alcohol and food consumption. A frequent inclusion was the requirement to serve food. And as at the turn of the 20th century, the rule became: whatever is cheapest, quickest, and saltiest.

Because OP asked about America, I didn't go into it here, but you might also be interested in my answer on drunk snacks in the Middle Ages.

You may want to x-post this over at /r/AskFoodHistorians

I just started reading Last Call: The Rise and Fall of Prohibition by Daniel Okrent and it was interesting to read that by the end of the 19th century and into the start of the 20th century, breweries owned a majority of saloons in the country. So much so that approximately 80% of saloons in the U.S. were owned by, or in debt to breweries. One of the ways saloons would attract patrons and keep them drinking (not because people were reluctant to visit a saloon but because competition was so fierce among saloons) was to offer free food. The book cites historian Jon M. Kingsdale who described a typical free lunch offered by a saloon in Chicago: "a choice of frankfurters, clams, egg sandwiches, potatoes, vegetables, cheeses, bread and several varieties of hot and cold meats." Another example in the book of food offered at saloons was an "assortment of sardines, pickles, pretzels and crackers guaranteed the one thing a hungry saloongoer could count on: the food would be so salty that only another schooner of suds could quell his thirst."