Here is another definition of flagship when referring to a “flagship university.”
“A flagship university tends to be the most known university in its state. It is generally the first public university ever founded and the one that receives the most support from the state government. It’s an institution people tend to search for the most, one with a good doctoral program profile. The majority of flagship universities count with and NCAA Division I athletics and are members of the Association of American Universities. Both groups are very important in the United States, being a part of them is a sign of leadership and prestige.”
I know the OP question is specifically about the UC system. But this question could be asked of other states. For example, UGA or GaTech. UGA is commonly considered the flagship and fits the definition of flagship university as it was the first founded. Based on the first definition of flagship, arguably GaTech could be considered the flagship as it is commonly considered the most prestigious.
As an east coaster with no connection to the UC system, if asked I would consider Berkeley the UC flagship simply based on reputation. No data to back that up! 