Someone will probably downvote you for making the irrelevant mistake saying US vs UK, but your point is absolutely correct that “America” is colloquially the name of the USA, not a continent or set of continents, in both US and UK English.
I suspect OP knows this, and is fighting a battle to try to change how Americans speak their own language — it’s not rare, in my experience, for people in other countries of the Americas to be annoyed by this usage.
I suspect OP knows this, and is fighting a battle to try to change how Americans speak their own language — it’s not rare, in my experience, for people in other countries of the Americas to be annoyed by this usage.