In United States, a telephone conversation usually starts like this.
Me: [Calls John]
John: Hello.
Me: Hi, John. It's Joe.
This is between 2 people that are familiar with each other, but not so familiar that they can identify each other by their voices. Hence, it is necessary to say who you are.
How is the last sentence translated into Mexican Spanish?
"Es José" is grammatically wrong. It seems like a word-by-word translation from English. I don't know if it is used somewhere, but here in Spain it sounds completely wrong.
"Me llamo José" could be used when calling an unknown person. Like if you are a sales rep or something, and you want to introduce yourself: "me llamo José y le llamo de la compañía X..."
"Soy José" is perfectly fine, and I would say it's the common way in Spain. This site says that "Habla José" is the polite way in Mexico, as Efren said too.
Number 2 is a common way.
A more formal way would be: Habla José
Joe talking. Joe talks is a one to one translation, not taking into account the verb conjugation.
In Venezuela, I learned to always respond with the form "Habla José," even with friends and family. I think that the formality of the statement will vary from country to country, but grammatically, at least, it's correct and fluent Spanish.
First you should greet the person picking up the phone.
"Good morning / afternoon / evening" - which ever applies
"Buenos días / Buenas tardes / Buenas noches"
Then you tell them who you are.
My name is José
Mi nombre es José
And then why you're calling.
