I had a similar thing, but on the phone. A woman called asking for the maid, Rachel (who used to look after my mom and now looks after myself, my bro and SIL!). I thought this was a bit unusual, but it could always be a friend of Rachel's who hadn't contacted her for a while and didn't have her cell number. So I said I would see if Rachel was home.
Go to her apartment, she isn't. So I tell the woman, "I'm sorry, but Rachel isn't here at the moment." The woman says, "What's your name?" I said, "You'll have to tell me what you want before I give my name." She says, "I need to speak to Rachel." I said, "She isn't here." THIS WOMAN KEPT ASKING FOR MY NAME, AND I KEPT SAYING I WOULD NOT GIVE IT UNTIL SHE STATED HER BUSINESS. Finally she said, "How are you related to Rachel?" (Related? Our ancestors came from completely different continents!) I put the phone down at that point.
The phone rings two minutes later. Bro answers. "We already told you Rachel is not here, do not call again." And he puts the phone down.
Thinking back, it was probably a debt collector or someone wanting to know about a store card Rachel has. Which is fine, but no way am I giving them MY name! AFAIK the woman did not call back. I admit that I thought at first she might she a friend of Rachel's because of her accent - she sounded like the same ethnic group. But when she started asking if I was related to her... obviously, she didn't know Rachel from a bar of soap. And trying to get MY name? Just weird.