Why is własny masculine but translated into "her own"?
WŁASNY refers to STYL, which is masculine.
You can say: "Ona ma swój własny styl", but it is not necessary to use the word "swój".
[I'm not sure if my polish sentence is grammatically correct] [I beg the natives to help me if I'm wrong].
It's perfectly fine. "swój własny" basically means the same as "własny", it gives more emphasis, but it's common.