It's basically a reflection of her identity conflict she has becoming Batwoman. She wants to serve as a soldier, but her only option is to become a vigilante. So the conflict between her Judaism and her LGBT lifestyle is a mirror of that in her civilian life. It's a battle between different parts of her sense of identity that they have used in the comics while in the show, Kate Kane is essentially Batman.
Look, her breaking into his cave and stealing his stuff is not what i consider okay. Especially since she made her own equipment based on her own personal identity and religious beliefs in the comics which is very empowering. However her taking up the name of Batwoman isn't really stealing who he is. He's still Batman and what the mask means to him is very different than what it means for her.
