identify buildings with a name different from the amenity/node it houses.
this is easily fixable by mapping POI as a separate object - as separate area or as a node
if POI is already mapped as a separate object then building should have its name, if any, not POI name (unless actually POI and building have the same name)
None of these seem to be part of the postal address, and hence should probably not go into an addr:-Tag, but into something else that Nominatim searches.
It’s possible that frequent values are sometimes wrong, especially as the feature isn’t directly used on www.openstreetmap.org. Luckily, there are 4900 unique values for 5600 uses.
The OSM definition is “The house (or building) name that is included in the address.”.
Often it can be a synonym for the housenumber, at least in places where buildings have only one housenumber.
If you look at https://www.oeschger.unibe.ch/index_ger.html you see that the “Uni Hauptgebäude” is not part of the address in that case, even though it is in the main building.
I would note it, but I guess you have to add building and office number when mailing them.
The postal routing analogy is a good rule of thumb and way of explaining it, but obviously has its limits and may not help determining what should be included in the search response. addr:floor would be an overkill. Personally, I don’t think “Länggasse, Stadtteil II, Bern-Mittelland administrative district, Bernese Mittelland administrative region, 3012” is that helpful, but I understand why it’s there.
Of the frequent values “Gemeindehaus”, “Pfarrhaus”, “Schützenhaus” can be helpful.