There is no official definition of what is a flagship university. In general, when people talk about flagship universities, they are referring to the most prominent public university of their state. It is usually the first public university that was established in the state and receives the most state support.
Gary Olson at The Chronicle of Higher Education offers the following definition:
While the criteria used to determine flagship status will vary from state to state, typically a state’s flagship is its land-grant institution. It is likely to be the university with the highest research profile and the most doctoral programs. It may house the state’s medical school, law school, or both. And it may be the largest and best endowed university in the state. Membership in the prestigious Association of American Universities may be yet another factor, and NCAA Division I athletics is a must.
The claim that the only “must” in the definition is NCAA Division 1 athletics may be more revealing about the value of flagship status then intended.
Depending on the definition, the flagship doesn’t have to be just one university. In Texas, both the University of Texas and Texas A&M are consider flagship universities and there is a push to elevate more universities to “flagship” status. However, the College Board appears to only recognize one flagship per state and in Texas that’s the University of Texas.
While there generally isn’t any official designation, there has been at least one case of official “undesignation.” The Idaho State Board of Education removed the word “flagship” as part of the University of Idaho’s mission because members felt it was unfair to other state universities.
(You can see one possible list of flagship universities in Why Your College List Should Start with Your Flagship University.)