In the wild west, before barbed wire fences, cowboys used to brand cattle so they could tell which ranch they belonged to. The definition of the word brand has expanded to include the identity of organizations like businesses, hospitals, churches, and universities. Organizations try to develop a strong or easily recognized brand so that people will choose them of of a crowd just like cowboys used brands to identify their cattle during a round up. An organizational brand has physical and mental dimensions. They usually use slogans and other descriptive words to form a mental definition.
Most people think of brand as a business term.
I think a better word for universities is reputation, people recognize it easily and have a high opinion of it. Reputation is a precious commodity. Just like with people, if you ruin your reputation, it's difficult to get it back. There are major advantages to having a strong reputation. First it's easier to recruit students.