No, it's not compulsory to stay in the hostel at GIBS Bangalore. If you prefer, you can stay in a nearby PG or rented flat—GIBS is flexible and even helps students find good accommodation outside the campus.
Also, GIBS is affiliated with Bangalore University for its BBA program, so the degree is officially recognized. For the PGDM program, it is approved by AICTE, which is the national body that approves top management programs in India. So both programs are well-recognized, just under different authorities.