Rice for my money has one of the two best residential college systems in the country (that is the usual name for that sort of system), the other being Yale’s. It is all based on the constituent college system of places like Oxford or Cambridge, and I think the only other real rival for those two in the US is actually the Claremont Colleges.
That said, once again WashU does have a version of that–their “South 40” dorms for first years are paired up into what they call Residential Communities, or in fact sometimes Residential Colleges, and they have events and IMs and such. But it is definitely not the lifelong association thing like at Yale, Rice, Oxbridge, or so on.
There are lots of others with some elements like that. Like I know Vanderbilt has something, and while I don’t really know the details, I gather it is a milder form similar to WashU’s.