grammaticality - "I have been Jessica" shouldn't it be "My name is Jessica"

We went to an electronics showroom, where we chatted with a sales girl. She explained some technical stuff about the things we were interested in. When she had finished explaining, she said

"By the way, I have been Jessica."

This usage is not new to me: it's pretty common in the US, and in Hollywood movies. But I have always wondered about how it could be correct (if indeed it is). I understand what it means, I just don't see it as grammatical. What is the logic of taking it as grammatical?

You have been Jessica. So what are you now?

A lot of people in the comments are saying that they never heard nor saw anyone using this line to tell their name.
Here is an example:

Apple Store Scene in Captain America, The Winter Soldier at 00:25

Answer

Although this usage is new to me, I think the formation of the idiom is fairly clear: It marks a (somewhat uneasy) merger of professional and personal registers as a sales technique. Thus, we wouldn't be surprised to hear a service person use a phrase like this: "I've been your flight attendant today," or "I've been your chef for the meal tonight." However, in a setting like the Apple Store, the simulation of a personal relationship with the clerk is a part of the service, thus the offering of the first name of the clerk.

Combined together, the idiom implies that the clerk offered you "friendship" but that it was just part of the service. "I've been (your buddy) 'Eric,' but now our relationship is over because you're leaving the store. But I can be 'Eric' for you again if you need me to be." This idiom represents the fact that the friendliness of the clerk is an acknowledged assumed persona. He may actually be named "Eric" but you are not interacting with the real Eric whom his friends and family know, but with a persona called "Eric." The use of the past tense releases both you and he from any obligation to think of the relationship between the two of you as persisting beyond the sales interaction.

In a cultural setting where people are encouraged to use personal relationships and traits for marketing purposes, it's probably a psychological necessity to establish some distance between the fake self and the real self. It also highlights how even constructions that seem obviously wrong can carry hidden meanings.