The structure for Pitt and Rutgers are the same in regards to hospitals.
Rutgers and RWJBarnabas are separate legal entities. RWJ University Hospital is the primary hospital for Rutgers RWJ Medical School, but they are separate entities, although they work closely together (and based on this announcement, Both Rutgers RWJ Medical School and Rutgers NJ Medical School will work closely with the entire RWJBarnabas network).
University of Pittsburgh and UPMC are also separate legal entities, although they work closely together.
Johns Hopkins is a similar structure, with Johns Hopkins University and Johns Hopkins Hospital being separate legal entities. They've formed a collaborative structure called "Hopkins Medicine", but they are separate entities.
University of Pennsylvania is similar with the university and hospital as separate entities, with a collaborative structure called "Penn Medicine".
I would not be surprised to eventually see the RWJBarnabas network collaborate under the umbrella of Rutgers Health. But since RWJBarnabas already has an established brand in NJ, I could see them continuing to brand separately, like Columbia and NY-Presbyterian.