I was late on the Batwoman series so if anyone can bring me up to speed as to why Kathy modeled her identity after Batman and why Batman hasn't shown up to tell her to stop I'd appreciate it.
I don't recall any in-canon explanation of how she got her look, just that her dad helped her get her gear together before they became estranged. Also Batman has shown up a couple of times now. Batwoman #0 was all about him investigating her trying to figure out who her secret identity was. After he found out it was Kate he tried to recruit her to Batman, Inc but she joined up with the DEO instead.