Can the word "customer" be used in all sorts of places (shops, organizations, etc) to refer to people who have gone there to run errands?

For example, a person who goes to a registry office to change their name. Or someone who goes to a hospital to donate blood. These people aren't paying anything because they are not buying anything. So when the person in charge of the hospital (or the registrars) wants to tell their employees that they have visitors, can they say, for example, "get up John, you have a customer"? Is there any more general and all-inclusive word for someone who attends a place to run errands but is not necessarily buying anything?

A person who goes to a registry office to change their name is a customer for that service. I would be surprised if no fee were involved, but regardless, a name change still involves a service paid for somehow, e.g., through taxes.

Someone who goes to a hospital to donate blood is a volunteer, not a customer in any normal sense of the word. They are not obtaining or benefitting from a service. They're also not a client, which, as per Lexico, is "a person or organization using the services of a lawyer or other professional person or company." Donating blood does not involve using any such service.