I don't think the distinction between country and city is meaningless. It's how the international system has operated at least since Woodrow Wilson and the doctrine of self-determination for nations. Arguably it has its roots much further back in the Delaration of Arbroath and other documents asserting national sovereignty dating back to the Middle Ages.
Even unionists who have opposed Scottish independence have not disputed that Scotland is a country and has the right to self-determination if it chooses. This is different than the situation in Spain for example where the Spanish state does not recognise the regions as having the right to choose self-determination.
You could argue back and forward whether it makes sense to construe Catalonia or Quebec or such places a countries or ever having been countries. However no one seriously disputes that Scotland is a country, not even the British state. Scotland was an independent country before it joined the union. Now it is a country within the union because it chooses to be. British prime ministers accept that Scotland could choose independence again if it wishes. They argue Scotland should not choose indepdence, not that it cannot choose independence. This is an extremely important distinction that Theresa May threatened to undermine by implying Scotland may not have a referendum even if it wishes. She didn't quite say that when she said "now is not the time", but it comes perilously close.
London is not a country. Comparing Scotland to London isn't just insulting, it's historically and politically ignorant.