This isn't true, it's a strange assumption often perpetuated online. Looking at cross-admit numbers actually published by Stanford's Faculty Senate, for example, Stanford's biggest overlap is with Harvard, Yale, MIT, and Princeton, in that order. It's a matter of pedigree - for undergrad, where grad school subject-specific rankings do not matter, there are an high concentration of very good STEM students at HYP; Harvard likely has the biggest concentration of excellent STEM students out of any university in America based on Olympiad winner matriculants and grad school outcomes, other than MIT. HYPSM lose virtually no students to universities other than between themselves, looking at the cross-admit numbers.

Schools like CMU, Berkeley, UIUC, etc. have different overlaps.

edit: This is just cold, hard data. Undergrads do not choose CMU/Berkeley/UIUC and whatnot over Harvard, Yale, or Princeton. Despite higher grad school rankings.