There is much discussion and indeed mission statements in the public sector about the need to focus on customers. However, I don't think the public services have customers in the same way that say Pizza Hut has customers. I think the public services are there to serve & engage citizens whilst being accountable - through politicians - to taxpayers/citizens. This makes the whole relationship a far more complex one.
I think saying that the public services have customers simplifies and indeed commodifies the relationship. For example if I want a pizza, I buy a pizza: the transaction has ended. However, if I, as a police officer (say), wish to serve and help my community become a safer place to live and work, I have to work with the local citizenry to achieve that. I cannot do it alone. There is no simple one way transaction there.
Yes, all public service workers should give good customer service to the citizens / service users / public that they are there to support and help. That almost goes without saying in my view. But that is but one small part of a much richer relationship.
The role of public service workers is to be transformational not merely transactional. The job of the public services is to generate sustainable social outcomes, not merely perform a series of one way transactions.
Sadly, the use of the word customer has become all to common in the public services. People may defend its use as reasonable shorthand. I take a very different view. If there is a shorthand to use, it should be 'citizen' not customer.
For me the biggest danger in this use of the word 'customer' is that it shapes (consciously or unconsciously) the business of providing care, education, community safety (etc.) into being a commercial enterprise. It is not. It is more subtle, more complex, indeed more important than that!