The way I have found best is to reverse the order of the facts. Whenever I have said, "Hello, this is whatsanenigma, may I please speak to so and so?" the conversation usually circles back and I am eventually asked "who is this?", whether it be that the person is going to transfer me to so and so, or take a message for so and so, or whatever the case might be.
What works better for me is to say "Hello, I would like to speak to so and so please, this is whatsanenigma." That just seems to be a better information flow for the one listening, and I don't encounter the circling of the conversation. It seems to be the "so and so" part that is important first-the person on the phone thinks about whether or not so and so is here-and then the name of the caller is next in importance-is this someone so and so knows? What is the relevance likely to be? Are the words "his daughter" included? And so on.
In the case of the OP, though, I don't think that the saying of her name would have done much good, if the person on the other end was so quick to assume it was a telemarketer. Many telemarketers will say "Hello, this is Mary, may I please speak to so and so?". They don't say "Hey, this is a telemarketer and I want to sell something to so and so."
And it doesn't seem like there was much time for the OP to get any clarifying information out, in that call. The man jumped quickly to a conclusion and didn't give her time.