This classic variation of the dish is based on the original recipe used by Alfredo in his Roman restaurant: nothing but fresh pasta, butter, and Parmigiano Reggiano. But yes, it's an American dish.
Making a good pecorino/parmesan emulsion is more difficult than you'd think. Cacio e Pepe is literally just pasta, pecorino, and cracked pepper but it's extremely difficult to keep it from ending up a goopy mess, short of silking it with cornstarch.
Check out Tasting History's video on YouTube about it. The Americanized recipe for chicken Alfredo and the like are typically cream based, but the original recipe didn't include it.
You can add more water from the pot and more Parmigiano Reggiano, the sauce is going to be godly. If you use "canned parmesan" tho, it's not going to taste good.