OSM doesn’t need to be confined to the exact implication. I say this because brand= is already used in education for a few university and college campuses. More governmental or public postal services viz USPS too, which is the 4th most numerous for this variation in https://taginfo.openstreetmap.org/keys/brand#values itself. Down the list, there is the British Heart Foundation, and the US’s Veterans of Foreign Wars. Then there’s the Co-op Funeralcare, and the Navy Federal Credit Union.
It is easily applicable to for-profit private “schools” especially kindergarten / nursery schools / daycare, as well as elderly homes and hospitals. If a charity have a chain of shops (eg Salvation Army), it would still use brand=, regardless of the underlying purpose.
There doesn’t need to another word for the same concept purely for such impressions and connotations. The UK government does many business case, and business plans. It has branding and brand guidelines. Doesn’t mean it’s a company. The essence is it has a recognizable common name component shown to the public.
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