Late to the party, but I will just throw out that while Chicago has a fairly distinctive vibe/branding, I think of the following schools as broadly similar: Harvard, Columbia, Penn, Johns Hopkins, and WUSTL. Basically, those are all private universities on the same basic model, and they are each the long-standing “top” such university in a major US city (Boston, New York, Philly, Baltimore, and St Louis respectively).
Incidentally, if you go back to, say, the 1900 Census, the largest six US cities were New York, Chicago, Philadelphia, St Louis, Boston, and Baltimore. Obviously things have changed, but I don’t think it is a coincidence that is the same list of cities with private universities like this.
Indeed, as various cities in, say, Texas, Southern California, and so on were rising to much higher relative size over the next 100+ years, the ecology of higher education in the United States was shifting, including that public universities were taking on an increasingly large role.
So, of course there are still important private universities like Rice and USC in Houston and Los Angeles respectively, and indeed you could include them on this list if you like. But I think there are some subtle distinctions in how universities like that fit into relatively “new” cities, including because of how universities like Texas or UCLA fit into such cities. Of course I know Texas is actually in Austin, but people I know from Houston seem to think of Texas as their top “home” university in the same way that, say, people I know from Detroit seem to think of Michigan as their top home university.
Anyway, that is my two cents on the universities I think of as broadly forming a “cousin” group with Chicago.