The US is a very informal society. I call my CEO by his first name. He calls me by mine. In many ways, it's assumed (outside of the job application process) that a first name is okay. My children's teachers write to me as "Mrs. X" in their first e-mail, but I consistently sign mine as "Barbara" so they use that by the second or third time. I've never said "Oh, please call me Barbara."
My husband has worked in retail (men's clothing) for a very long time. He thinks anything that shows you consider that person to be a real person instead of a non-feeling robot is a good thing. He always uses the person's name if that person is wearing a nametag and says things like "Thank you, Mary, you've been very helpful today" or will joke with the person and say "Everyone on the street says Peter is the guy who will take good care of you." Most of the time, when you say things like that, "Peter" will indeed start taking a more active interest in helping you because you've shown interest in them as a person.
Very rarely, when I get a bad feeling about someone (they are tyring to sell me something I don't want on the phone, or whatever) if they say "May I speak to Barbara X" I will say "Yes, this is Mrs. X" instead of "Yes, this is she." Using Mrs. X. gives a signal that I do not consider them my friend and I do not want a personal relationship with them. This is rare.
I cannot imagine ever saying to someone "Please call me Mrs. X" if they call me Barbara.
An exception to all of this, of course, is children and adults. My children's friends all call me Mrs. X.