Wowser, what a fun question!

First we can eliminate the impossible: It definitely wasn't Alfredo as that actually came around much, much later than those lyrics. It also wasn't bolognese as that, too, arrived a bit after your song was applied as a generalization of those backwards bumpkin Americans, though it did at least come about in the right century (1700s). Also, generally speaking meat was relatively scarce in the vast majority of all Italian^1 cooking and especially prior to the 1800s, as evidenced, for example, by bolognese appearing first at the end of the 1700s. Marinara was around at the time and had been for a while already, since shortly after the conquistadors returned with tomatoes to Europe in the 1600s. That dish started in the southern region as a simple and common dish believed to be enjoyed primarily by sailors, as suggested by its name (some sources note the origin instead to be that it was first used on seafood which is not nearly as likely to be the origin... regardless of who is correct it doesnt impact the origin location or timeline either way and so is ultimately moot to us here). Anyhow, there is no good reason to suspect marinara was the dish in question either as that is quite literally a sauce and calling a specific sauce dish "macaroni" just wouldn't make much sense at all given what the term macaroni meant. But there were other common dishes that we don't see today on the menus in Fabio's and/or Sal's in our respective towns (I'm 99% certain every American town has at least one Italian joint that has one of those two names), so what about those dishes? What actually was it then?

You've said macaroni is "just a shape of Italian pasta that could be used with any variety of sauces and dishes," and you're probably more correct than you think you are in that phrasing. Macaroni, in the general and common American vernacular usage today, of course just refers specifically to elbow pasta. While that is indeed a versatile pasta style that can, in fact, be used in a variety of dishes, that's not really what the word meant then and there and would be incorrectly applied to that time when using our modern common American definition. Maccheroni, from which we derived the term macaroni, was much more broadly applied than just referencing elbow pasta. In Italy it actually (historically) means any tubular pasta of a relatively common size diameter regardless of shape or length, and in some instances was used even more broadly within the larger pasta category. So it wasn't anything specific like it is to us today, it was instead a larger subcategory of similar pasta types, however that general style of pasta was relatively commonly served and very simply, and it also helped to create the style of the common Italian kitchen which continued for several hundred years and in some respects still does today. For instance, real Alfredo (developed/popularized in the early 20th century) has only three ingredients: butter, parmesan cheese, and pepper. Though it's a relatively modern dish it is in keeping with the authentic traditional style and simplicity of Italian cooking, and maccheroni as a dish is a very early traditional example of that style, colloquially being a very basic dish of a macaroni style noodle combined with butter and parmesan cheese (which parmesan itself dates back to 13th century in Italy). This recipe combination dates at least to the 14th century, actually, and then it slowly spread outward across Europe from there and was ever so slightly modified. So they were indirectly talking about, in essence, macaroni and cheese in that song - though, depending on where they had it, it wouldn't be very much like like typical American mac and cheese found today. And remember it wasn't simply enjoying the dish itself that earned the moniker, it was effectively referencing (and mocking) those individuals overly embracing of foreign culture, dressing overly ornate yet also somewhat out of current high English fashion, acting a bit too effeminate, wearing high powdered wigs, and generally defying the established behavioral customs of the English elite.

Wait, did you really think that was all I have to say here??? Well, my friend, this is AskHistorians. While that pretty much answers what you've asked about the specific dish that was later used to describe those backwoods outdated-fashion-style-sportin' dumb-dumb Americans it doesn't quite explain the whole thing... and, well, that's kinda our entire purpose for existing here, or whatever, so let's go even deeper into some history;

As you've already noted, many British (and fancy pants type) young men toured Europe to become cultured and experienced, particularly in the early to mid 18th century, with quite a few of them getting a little too enamored with their journey and being mocked for it due to their behavior and demeanor. This rite of passage type journey, in the larger context, came during a time of huge societal transition between old ways and new ones, while old theories and religious decrees were attacked by new science as revolitionary new political and religious concepts found strongholds, and the travelers helped fuel the cultural spread that was growing like a wildfire at the time. Galileo Galilee built his telescope in 1608. By the 1660s ice houses (buildings used to store frozen blocks of ice typically taken from rivers or ponds during winter) started to become common (for wealthy estates) particularly in France, England, and Italy. In the 1680s Isaac Newton published his Philosophiae Naturalis Principia Mathematica about how falling apples keep falling until they hit you, or hit a squirrel or something, or else they'd just fall forever and ever, also with other whatnot about objects, motion, forces, and their rules. Pretty much immediately after that was published John Locke published his Two Treatises on Government (1689) from which came revolutionary and much elaborated upon concepts like an unwritten default social contract inherently existing respecting and protecting rights of all its members who were bound to the contract just by participating within society, further explaining that these rights also naturally included Life, Liberty, and the recognition and security of personal property, and with the authority of the collective society being entrusted to its leaders by consent of the governed - those fundamental concepts so eloquently expressed in the Declaration of Independence and in many of our state constitutions less than a century later. It was a quickly changing world in which new North American Colonies had been successfully established and stabalized, providing vast resources and trade wealth to England, as had happened for Spain in the prior century (though initially they were not nearly as profitable as Spain's had been). Also during the 1600s Italian cooking was continuing to develop in complexity while French cooking began to really develop as a competing style that even started to match the quality of the ideas played/playing out in Italy... in all of this development into the next societal chapter the exchange of cultures and foods were happening, too, with new foods coming in from far away lands, cooking styles being further developed and refined as improved cooking methods spread, and folks also gained more access to more devices and tools.

By the early 1700s the elite leaders within British American Colonies were emulating the latest London trends, which in turn had come from all over Europe in the London elite's attempt to be on the top of the current cool, well, everythings. One for instance is dancing. Prior to the 1600s few publications on dancing existed, and almost all that did exist dealt with proper dancing specifically in the court of monarchs. In the last two decades of the 16th century some new publications covered new dances but this was also during the time of Pilgrim-type religious devotion (i.e. not playing, dancing, etc) and soon came rebuttals like Traitté contre les danses, a French publication from the very early 1600s declaring basically all dancing as a sin against God. Regardless of those protests, by the end of the 17th century numerous books existed covering men's solo, women's solo, duet, and even group dances and it had become common all over England. Dance master Raoul-Auger Feuillet developed new dances in the latter half of the century which quickly spread elsewhere. By the early 18th century dances and balls were even being thrown by Alexander Spotswood at the Governor's Palace in Williamsburg, Virginia, and within the first quarter of that century the first playhouse would open in an american colony (also in Williamsburg). Another thing that spread with this desire to embrace high culture was food from Europe.

Cont'd