I have used a couple of approaches to the idea of anonymity.
I think your best option is to emphasize the value of the non-anonymous survey response. My experience is that customers concerned about anonymity wory about reprisial or negative impacts if they provide candid feedback. As a researcher/company you want to cultivate a relationship where your customers feel safe providing the most cadid feedback they have. You can explain that you keep the personalized information so that you can follow up on concerns and complaints and address or resolve the customer's concerns.
Another way to address the concern is to explain that while you retain personalized information within individual responses, survey resuts are reported and analyzed in groups which minimizes the feedback of the individual in lieu of examining the feedback of the group in aggregate. Essentially the difference between explicit anonymity versus functional anonymity. This is a less ideal solution for a number of reasons. It minimizes the value of candid, detailed responses and makes the survey relationship feel more cold and detached. That said, it can directly ease the concern that the customer will be isolated and targeted because of what they said.
My organization follows up directly with detractors of one of our surveys to see what, if anything, we can do to address the situation that caused their negativity. Without the personal information we retain with each survey we could not provide that personalized touchpoint.