Do you mean "call someone on the phone"? If so then people will say "This is Mary" or, more commonly, "It's Mary" as long as the person is likely to know you as Mary. If you don't know each other then people will commonly say "This is Mrs Surname speaking."
To introduce yourself on the phone by saying "I am Mary" would seem distinctly odd to me although if they don't know you you might say "My name is Mary" and then say why you're calling them.
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