Just wanted to say, Haverford is not down the list from Carleton in terms of prestige. Both schools are usually in the latter half of the USNews top 10. They are peers, IMO.
Regarding Yale:
It seems there isn’t quite a Yale doppelgänger yet, a school that is highly similar in all main aspects (academics, social vibe, environment, dorms, etc.):
Academic rep/emphasis:
Yale is known most (academically) for its Humanities, so a school like Wesleyan, Kenyon, Oberlin or Vassar – as mentioned previously – would fit the bill there.
Academic breadth/quality:
UChicago, yes, in terms of overall breadth of program quality. (and limited Engineering)
Social vibe:
Yale is liberal but also has a mix of preppy/artsy/nerdy – any number of schools have a similar makeup.
Housing:
Yale has the aforementioned residential colleges. Other schools that have them, like Rice, have been mentioned.
Look and feel of campus: Gothic, grandly so. UChicago, Princeton, Washington U and many others have that style of architecture.
Environment: What places are similar to New Haven in terms of size, feel, cultural opportunities?
Anyway, I clearly am having a hard time finding a school like Yale in all of these key facets. Maybe if you decide which things are most important, we could aim our casts at those parts of the lake. (and if you already have, I apologize for missing it)