The rankings from many sites are set up to provide ranking that matches the perceived prestige of the colleges. The people who create the ranking methodology look at colleges that they (the creators) perceive as “prestigious”, and they develop a methodology based on what these colleges do better than other colleges.
In many ways, most are not that different than “Hunt’s prestigiosity rankings”, mentioned by @Data10 above. They are simply methods to rank a college based on how similar this college is to the colleges that they think are the most prestigious.
As has been mentioned many times before, college “prestige” is a beauty contest, with the standards of “Beauty” generally determined by the influential (the wealthy and powerful), but with assorted groups having different standards of beauty.
Overall, the wealthy and powerful in the USA have decided that the Ivies+ (especially HYP) are the standard by which all other colleges should be measured, and demonstrate this by preferring these colleges for their kids above all others. Ultimately, that is the true measure of “prestige” among the general college-bound public who care about this prestige.
Prestige is a metric which is based on the average amount of power and influence held by the families of the students who attend the college modified the amount of resources and time that people are willing to invest so that their kids will be able to attend these colleges.
Harvard is full of the kids of the wealthiest and most powerful, and families are willing to put tens of thousands to millions and years into making certain that their kids will be able to attend Harvard. Families donate millions to ensure that their kid will be accepted, pay hundreds of thousands for consultants and for full pay. Kids and their parents are thinking about how to be accepted to “prestigious” colleges before they start high school, and spend their entire high school focused in ways to increase their chances t be accepted to a prestigious college. The $500,000 College Consultants often start working with the kid from the kid’s freshman year. Some families are so obsessed with their kid attending HYP that they decide which kindergarten to send their kid to, based on which kindergarten will feed into the elementary school that feeds into the elite high school, which a feeder into Harvard (or Yale, etc).
My point is that the “prestige” of a college is, in many ways, the result of the effort and money put into being accepted to the college, rather than just being the cause.
In other words, I think that “prestige” is more the result of popularity, especially among the wealthy and powerful, than it is the cause of popularity.
PS. I’m talking about general prestige among college-bound families. There are different groups that have their own definitions and determinants of prestige, like faculty, people who are mostly interested in specific majors, etc. I’ll also agree that sometimes, prestige and popularity is based on local connections, like state flagships which are generally popular in a state because they are state flagships (The Ohio State is popular in Ohio and both its prestige in the state and its popularity in the state derive from this fact, rather than from each other).