Some languages have something between a singular first person and a plural first person, called a grammatical dual.
To quote from the article,
Many languages make a distinction between singular and plural:
English, for example, distinguishes between man and men, or house and
houses. In some languages, in addition to such singular and plural
forms, there is also a dual form, which is used when exactly two
people or things are meant.
When a nurse asks "how are we doing," she may be using the grammatical dual form. As the immediate meaning ("How are you and I doing?") makes little sense, the patient immediately understands this to be an indirect way of asking how the patient is doing.
(I read this in a Mad magazine in my misspent youth, I think. "Nurse: How do we feel?" Annoyed patient: "Half and half. I feel fine, but you look awful.")