«Personally, when traveling abroad, if asked where I'm from, I generally say "Washington DC", not "the US".»
Yeah, I've met people like you. The problem is, this asks from your interlocutor of awareness for internal USA geography knowledge. On the other hand, many of those "from San Francisco" or from somewhere alike expect you to be just "from Europe" instead of "Germany" (or any other national state with its own cultural identity and external policy, for that matter).
P.S.: Try "Franconia" or "Thuringia" instead of "Germany" and see how many bells that rings! Mind you, these are broad regions, not merely cities.
It doesn't need any extra knowledge, assuming people know Washington DC is in the USA. They're no worse off than if he says "the USA".
I often say "London". I don't like saying "England", I consider the whole UK to by my country, but "United Kingdom" or even "Britain" are often not understood, and in any case, "which city?" is almost always the next question.
There are exceptions. If I'm introducing myself to someone with very limited English, like a young child in a developing country, I'll say England.
It happened to me quite a lot to introduce foreign clients to local acquaintances that haven't traveled abroad and don't have much knowledge (or interest) about outside world. Yeah, they heard of Washington (DC), alright, but haven't heard of SF and that caused a scene to my local pals with them being clearly unaware of where that place was and also being a little embarrassed to ask more. US citizens are not the only ones, this also happened with Vancouver and Melbourne. These are indeed prominent cities but not necessarily in anyone's world, and it seems nicer when that "which city?" comes instead of "where is that?"
It's at least a bit dependent on who is asking. I would expect somebody from Western Europe, Australia, etc to know Washington DC and SF (and London, Manchester, Frankfurt, etc).
I wouldn't have the same expectation of somebody from Asia. Not only is language a possible problem, but there are a TON of large cities in China (and other parts of Asia) I know nothing about.
Vancouver and Melbourne I can understand, but I'm amazed that there are people in the developed world that haven't heard of San Francisco.
That city has been part of the American "brand" for decades; from hippies, free love, Vietnam protesters and gay rights, to the dot-com boom and more recent startups. Add some destructive earthquakes, plenty of American films and music, and the international press whenever a startup does something bad.
Some (many?) Americans would describe themselves as hailing from a specific state or city, not the nation.
Personally, when traveling abroad, if asked where I'm from, I generally say "Washington DC", not "the US".