There is a saying that imitation is the best compliment. But your sister-in-law’s behavior suggests something serious. Taping your conversations goes well beyond flattery. I think you are correct to worry about how far this will go.
Although a certain amount of copying others is normal — that’s after all where fashion trends come from — this level of excess is not normal. It speaks to a kind of desperation. Not knowing who she is or maybe not liking who she is, she cloaks herself in your identity. I can only begin to imagine how empty she must feel to go to the lengths she does to become your “twin”. It must be frustrating indeed for you to feel like she is intruding on your self-expression.
I’m sorry her husband won’t deal with it, although I can understand that he may just want to minimize drama in his relationship with her. It may be that he is already making many concessions to live with her. I do think that loving her means getting her some help for how bad she feels. A therapist would help her develop a stronger sense of self so she would feel more comfortable in her own skin.
As for you: Forget about “confronting” her. It just feeds into the problem. Ideally, distance yourself as much from her as possible. Don’t make any big announcement about it. That will only trigger her. Just quietly try not to see her so often. Switch things up whenever you feel like it. Try out different churches. Experiment with different styles. Don’t tell her in advance. Just do what you want to do. Above all: Compliment her on anything she does that is unique to her, and encourage your brother and other members of the family to do the same.
You might want to see a therapist yourself — not because you own the problem but because a therapist might have suggestions that are more specific to your situation than I can offer.
I wish you well.