Yeah, this is a power play, a game to this woman. It's her way of trying to make you uncomfortable and make herself look knowledgeable. She is probably short on self esteem and tries to gain an upper hand in conversation to make herself feel better.
Information is gold to her. It's used to make her seem important.
It would be best to be BIFF with her, Brief, Informative, friendly, and Firm.
Don't give her personal information or extra information.
You were right to agree with her when you can. If she could cause a division and triangulate people she would. Be cool as a cucumber. Do not reflect her emotion. She won't know what to do if she can't fluster you.