I don’t look how many building=school polygons exist but how many different amenity=school nodes, because multiple of the amenities may be within the same building, and not each one on its own.
In any case, I add the address in the amenity only. In the building I would add the address only if it’s a named building, usually a listed building which may also have a Wikidata item.
I suspect the answer might need to depend on the country. Different countries assign addresses to different things in general. That can affect user expectations and mapper tendencies.
You are putting a lot of emphasis on buildings with these questions, but in not so few places, addresses do not even refer to buildings, e.g. in Italy they refer to entrances and gates (and it is likely a school has several of them, but often will use a “canonical” address which is one of them), in many parts of Germany as well, addresses refer to plots and not to buildings.
That’s more or less what I see in my part of Spain too. Entrance nodes usually have the number and street name but not the school name. The amenity=school object usually has the full address including the school name and the number of (one of) the entrances.
I think there is nothing special about schools here - it’s similar to how other POIs are mapped locally.
I don’t see many address tags on buildings here except where a building and a POI (amenity, shop etc.) use the same polygon. That is rare for schools as there are usually some grounds outside the building.
@Mateusz_Konieczny The questions refer to the two simple mentioned situations without any caveats. I’m well aware that the wide world offers almost unlimited amount of edge cases. That can be seen from the 2 replies after yours :).
@dieterdreist, yes I’m putting intentional emphasis on what look like the few most common cases since I’m mainly interested in them. Thanks for the information though.
Point of my answer is that you need some caveats to make it answerable in general.
Question likely makes sense if narrowed down to set of areas including Bulgaria, but not as general and worldwide. I mentioned it mostly to discourage using answers in such way, for example by adding them to OSM Wiki as a general advise.