I think you might be missing the point a bit. It's not that you need to hide the fact that you're an American. It's just that (in my opinion) Europeans (including the British and Irish) are slightly more formal in their dress, and wearing (for example) a red, white, and blue track suit with new white sneakers will suggest that you are a tourist and a somewhat culturally insensitive one at that.

It's not an "American" thing necessarily. I can sometimes cringe about a group of German tourists in my travels.

But you sort of brought on the responses you're commenting about when you asked, "I'm going to London and Ireland; how should I dress?"