Blair House is... a house. A family named Blair used to live in it. So, not really parallel to the British usage of "House" for structures like the BBC headquarters, "Broadcasting House", which have never had residential usage, and are in no way suited for it.
Sure, I can name lots of structures in the US that have names, but they are far from the majority, and I an unaware of any legal weight for the name (IDK if UK building names have legal weight either though...)