Is 'Confirm' the most appropriate word?
Solution 1:
Good question!
Of course if, at the start of the conversation, you have mentioned your name, there is nothing inappropriate about their asking you to "confirm" it. However I do agree that people often ask for "confirmation" of something about which they currently know nothing.
Were it me asking, I would probably say something like "Ok, we can deal with that, but first would you mind giving me your name and address, please"; or "may I ask your name and address, please".
Solution 2:
If you are speaking to a company with which you have previously done business, the company representative is asking you to confirm that the facts recorded in their database (which the representative is probably looking at on a monitor) are correct. It is always possible that the facts were entered incorrectly, or that they have changed since your last encounter—you may have moved, or gotten a new phone number, even changed your name.
And the person you are speaking with probably handles fifty or a hundred such calls every day, mostly with existing customers. Under these circumstances their language quickly becomes stereotyped: the form of a question is triggered not by the circumstances of the call but by the particular datum they're seeking, so they're quite likely to ask you to "confirm" a datum even if this is your first contact with the company.
If I were the rep, I hope I would say something along the lines of "Let's start with the paperwork — could I have your full name, please?" But as the caller I wouldn't be particularly bothered by being asked to "confirm" it.