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

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.

Like others, I've never see that construction. But I remember listening to the BBC World Service a lot in the late 90s and the reporters were often signing off with "This has been _________." Although that sounded a lot more natural, at some point different reporters were joking about it themselves, joking that it made them has-beens.

Unless it's some kind of deliberate, specialised usage (a company's quirky greeting), I would say it's a mistake and to be avoided.