A few data points: There are 5 students from Harvard in my year at Stanford. MIT last year accepted multiple students from several universities in the first round. Columbia, if I remember correctly, did not make first-round offers to multiple students from the same university. I have heard from a grad student at Penn that a classmate of his got waitlisted at Penn and was rejected on the same day that he accepted Penn's offer.
Admission procedures seem to vary by graduate program.
It is certainly true that applicants from the same university can be more easily compared side-by-side - especially if you also have overlapping references. That same phenomenon is a huge concern when two grad students with the same advisor finish together and apply for academic jobs at the same time. Benevolent advisers will tell half of the jobs that student A is the stronger candidate and the other half that student B is stronger. Otherwise the stronger student would end up with all of the job offers and the weaker one might be left empty-handed. Or worse, the advisor might try his best to give both students "equal" recommendations and then they both sound fishy enough that neither student is successful on the job market.