AITA for not giving my name to a customer over the phone?

I work at a store that sells electronics, whiteware, TVs etc. We are a large company with several stores throughout the country.

On Friday, the phone rang, so I answered. The conversation went as follows:

Me: "Company name, town name, can I help you?"

Customer: "Who am I speaking to?"

Me: "Company name, town name." (In a pleasant voice.)

Customer: "Well if you're not going to give me your name, I'll deal with someone else!"

Hangs up on me.

Today, (Monday where I live), my boss received a complaint from our head office about it!

He (the customer) said it was the rudest service he has ever had!

I was shocked. I don't prefer to give my name, because in today's world of social media and online reviews, a customer can name you in a complaint online and we are not allowed to defend ourselves or get the comment removed. It's there forever. I have had this happen to me, and it is incredibly distressing. I realize some complaints are justified, but some are not, and it makes no difference. The customer can say whatever they like.

If the customer had asked again, specifically for my name, I would have given it, it's just not something I offer out if I can help it. (We have name badges, so it's unavoidable people knowing who we are, but I'm just not comfortable with it.)

Am I the asshole?

Info: There's no company policy at all on how to answer the phone or what information to give out.

Info: Because live in a small town, having my name in a bad review online is distressing and humiliating, even though it's only my first name, everyone knows exactly who I am.