In Poland Lidl, Żabka, IKEA, Awiteks, Good Lood and others do not have unique local names between shops.
I am pretty sure it is true also across Germany, Croatia, Ukraine and several other countries I visited.
In fact I expect it to be true in general.
Note: they may have unique internal names. And chains obviously distinguish by address and location. And likely some chains have shops with unique names per location.
Search results on website may show such location description is view title. Such titles are NOT used by people as common primary name and do not go into name tag.
No, it is about a general consensus. If you try putting ref into name I bet that you will get strong opposition.
You are pushing some pet theory mismatching actual tagging in OSM.
Different supermarkets or convenience shops of the same chain do not have distinct primary names.
name tag is for commonly used primary name. If philosophical “proper name” gets distinct results then it should not be used for describing name tag.
If your intention of pushing “proper name” was to claim that each shop of given brand should have unique name tag set then I think we found critical issue with your proposed terminology change.