This is a little... If not necessarily incorrect, then at least odd. If the conversation literally does just go "Oh, your accent, you're American huh? Are you married? Give me your number." then, you've definitely got a fair gripe, but given how often this seems to be happening I'd stick my neck out and say that this isn't quite the case. If it isn't and it really is just a normal-ish if clumsy attempt at hitting on you, it seems to be quite harsh both to the people you meet and to yourself. First of all, I think it's almost certain that this happens because it's a combination of finding Americans interesting, attractive and (stereotypically) friendly. If it goes too far - that's a big problem, but it's a different problem and not really to do with your American-ness as much as it's to do with the fact that these guys are rude and insensitive. If they aren't being like that but it just is because they're showing a normal level of interest, then again it isn't especially their issue. Perhaps you just don't like how Londoners behave, but again, that doesn't really have anything to do with a prospective green card. It's a little insulting to assume that all these people walk around disliking London and Britain so much that the second they meet an American they view her as nothing more but a ticket out.
As has been said, it's also an easy way to keep a conversation going and find out about someone. It also makes you a little more individual, in a city which is otherwise quite anonymous if you're just run of the mill. Also, probably you're just a friendly, attractive person in and of yourself? I mean, I don't want to tread on your toes, but why assume that they want you for your green card when really, they fancy you? I'd take a bet and say you're not so hideous and horrible that your citizenship is the only thing you've got going for you.
I think you're trying to read your being American as the *only* aspect to people's interactions with you when, really, that's not the case. You may stick out for being American, but no-one's going to let that fact dictate everything.