I'd like to think that I would have laughed off my brother's remark as a joke and replied, "Yup, I guess I am," with a genuinely confused expression, then beandipped. But of course I don't really know if I would have since it didn't happen to me.
Regarding the OP using a pronoun instead of Mrs. Donny's name when referring to her:
I wouldn't expect any English-speaking adult to have constructed their sentences differently.
I don't know if this is the type of thing that only bothers writers, but it drives me nuts when in the first few scenes of a play or the first episode of a TV show, the characters repeatedly and artificially refer to each other by name when a pronoun would suffice. It's bad writing because normal people don't talk that way.
I get that they are introducing new characters to us and want us to remember their names, but it comes off as stilted and fake. People don't say, "Mary, did Jennie tell you about her school trip? Jennie goes on lots of school trips. On Jennie's last trip, Jennie bought a kite then Jennie brought it home and we went to the park."
They would use "Jennie" in the first sentence, then use "she" and "her" afterwards.
In the OP's case, she referred to Mrs. Donny as "she" because that's how normal people talk. She was a real person in a real-life situation, not an actor in a poorly-written sitcom.
She'd already mentioned Mrs. Donny by name when politely introducing the topic of where she was from to her brother as a point of conversation. Therefore, in every sentence following that pertained to that discussion, her brother knew who she meant. So, to turn to him and say, " I don't know why Mrs. Donny said that. Mrs. Donny is from there!" would have been odd.
If, in a conscious attempt to not use "she," the OP had instead turned to Mrs. Donny directly and said, "I don't know why you said that. You are from there!" while the woman looked down at her plate, that would have been far worse, IMO, and very confrontational.
Maybe people are just making a point that they feel she shouldn't have said anything about Mrs. Donny at all, though.
The OP couldn't have known that discussing a subject Mrs. Donny seemed to have taken pride in before would have upset her that time, so I think she was understandably caught off-guard when responding to her brother's "accusation" of her being a liar.
(I still hope that he was just kidding and flustered, though.)