Yes, there is a we that is only used for another person (or people). It is sometimes called the "patronizing we" (example), the "all-inclusive we" (example), or the "nurse's we" (example). One paper says the following:
Moreover, the patronizing we or all-inclusive we can be used in addressing instead of you. A doctor may use this to give hope to patients or to indicate that he is part of the situation.
Making health communication accessible: A rhetorical analysis of radio health talk (see page 61)
The corresponding definition in the OED is:
Used confidentially or humorously to mean the person or persons addressed, with whose interests the speaker thus identifies himself or herself (esp. by a doctor in friendly or cheering address to a patient); also used mockingly or reproachfully by a parent, intimate friend, etc.
Paywalled link to OED
For example, in this Monty Python sketch:
Doctor: Morning, Mr Henson ... How are we today?
The earliest citation in the OED for this sense is 1702, so it is not as old as the "Royal we" which dates back to Old English. Just for completeness' sake, the 1702 quote is this one:
Well, old Acquaintance, we are going to be Married then?