I think there is a tendency in Bulgaria not to use names of squares in addresses, but it does happen, at пл. Възраждане for instance (Way: 1 пл. Възраждане (275704547) | OpenStreetMap). Here the square is used in addresses of the buildings adjacent to it, and the streets in front of them are named with addr:street=пл. Възраждане (not addr:place=пл. Възраждане, I think we should use addr:place=* only for neighbourhoods, in combination with addresses consisting of block numbers).
At пл. Жърналист, however, the name of the square is avoided in the addresses of the buildings adjacent to it, so here I think the square should only be mapped as place=square (preferably as an area) + name=пл. Жърналист.
On пл. Св. Александър Невски, the only building using that name in its address is the cathedral (according to cadastre, without house number). Still I think that this is enough reason to add name=пл. Св. Александър Невски to the streets surrounding it, if they don’t have an obvious other name (indicated by buildings using that other name in their addresses). I also understand that routers often have a problem with understanding named areas (that’s why we are mapping paths on the main diagonals of a pedestrian area, even though they are not visible as separate paths on the ground). So that’s another reason to add names to the roads on пл. Св. Александър Невски. I don’t think having a name on both the streets and the place=square area is a problem.