Because MAHE is NOT a private university but a deemed university and according to UGC regulations, a deemed university can’t have “University” in it’s name. That’s why SRM University also changed it’s name to SRM Institute of Science & Technology.
On the other hand, Manipal University Jaipur IS a state private university, which is UGC recognised. So Manipal University Jaipur can have “University” in it’s name.