I would be surprised if any customer service gave a real name over the phone rather than an assigned nickname or badge number.
When I worked in a help desk (working directly with customers over the phone) for an ISP, I always just used my real (first) name. The help desk was (still is) small enough that if there was ever a complaint/compliment that would be enough for that feedback to get to the right representative. I don't know how common that is, but I think a lot of places (particularly if it's US-based) will just have people use their real first names unless that particular rep would rather not do so.