I don't know my policy number

Today I called Geico, who I've been very happy with, to pay a bill. As soon as someone picked up on their end the first thing they said was, "Thank you for calling Geico, what's your policy number?" First of all, I don't know my mom's phone number, you think I know my a twenty five digit alpha numeric account number?

Second, is that anyway to answer a phone from a customer?

Why do large company's instruct call centers to answer the phone this way? I understand time is money but a few second would really help build some rapport, especially when 99% of my interaction with your organization is when I'm paying a bill. Wouldn't it go a long way to start off like this:
"Thank you for calling Geico, my name is Winston Churchill, may I ask yours?"
"My name is Jeff Brainard."
"Oh, nice to meet you, what can I do for you today?"
"I need to pay a bill."
"Okay, great. To look up your account may I ask your phone number sir?"

Today the lady was very sweet but she couldn't find my account because I didn't have my policy number. Then after ten minutes of confusion when it didn't pull up with my SS# followed by placing me on hold she asked me my phone number. Within a second she had access to both of my accounts. If you can find a person's account using their phone number, why bother asking them their policy number?

When you begin a conversation with a customer with a question that they don't know the answer to, things don't usually progress in a friendly manor. By doing this you put them on the defensive, as if they've done something wrong by not knowing their account number. Additionally, when a customer calls and they have a generic question being asked to eight questions to verify their identity is increasingly frustrating. I've called companies to find an address of where to mail a bill and been put through a mental triathlon remembering validating personal information simply for me to get a PO Box that would be the same PO Box regardless of my mother's maiden name.