“Became” is when it happened in the past only!
Ex 1: “I lost in the market” true has serious consequences in the present, but the “losing” process has stopped in the past!
Ex 2: “the name of the team became a symbol of a fast and furious play” means it once became a symbol in the past but is not a symbol of a fast and furious play anymore.
“the name of the team has become a symbol of a fast and furious play” means in the past it became a symbol and it’s still a symbol.
If the teacher said you should say “has become” it’s because symbols of something often are for good, once they’re a symbol of something, they can’t stop being that (except maybe in very rare cases or so, in which you use became instead of has become, depending on the context)