My POV as a Legal Alien:
Building on what others have said, the biggest mistake anyone can make is assuming that just because the British and Americans speak the same language, we possess a much greater degree of cultural similarity than other nations. We don’t. The fact that we share the same first language (and can therefore express and explain our differences to each other with a great deal of complexity) does mean, however, that it is possible to explain, understand and overcome our cultural differences far faster and to a greater degree of certainty than any interaction between cultures that don’t share a primary language.
Its always worth remembering that a lot of our cultural identity and understanding comes from shared or similar experiences with others. From a social perspective, most of those shared experiences happen at a time when we are all living broadly similar lives to our peers - childhood and early adulthood. The upbringing of your average British kid is a world apart from that of your average American kid - the tv is different, the education system is different, the memorable times and historic events are also different - or at least carry different importances and perspectives. We even “come of age” in completely different ways and at different times in our lives. On top of this, the consumer culture and big brands in America are very different from those in Britain - sure people like Kellogs are global, but the majority of the small stuff that you recognise from home (in either country) cannot be found in the other.
Why does all this matter? Well these are all things that have had a major role in informing our beliefs (and predjudices) and providing the context and framework in which we exist. Next time you are talking to friends or are at a social gathering*, make a note of just how many conversations, arguments and jokes rely on an implicit understanding of a concept or physical object that relates to something in the above paragraph. You’ll also find that most of those that don’t still rely on knowledge of a long-passed event, national personality or tv programme that never left the national shores (again, this applies to both countries).
Speaking as a toast-done-on-one-side eating, you-drink-coffee-i-drink-tea-my-dear Englishman exiled to these barbarian shores i can honestly say that the biggest problem i have here is not understanding what people are saying, but understanding what people mean.** The fact that i’m saying that as a Londonner living in New York speaks volumes, because there are no other two cities in the world who can claim to be so geographically and culturally distant and yet so similar on so many fundamental levels.
I feel my most alien in a social setting, because i have a tremendously difficult time relating the experiences and culturally-linked metaphors of Joe American to those of Johnny British and i have to think very carefully about my own comments and vocabulary in conversations to make sure that the Septics i’m speaking to don’t have the same problem.***
That’s not to say the vast majority of those differences can’t be overcome, however. It just takes time and cultural context. There’s also, as has been pointed out, a difference between “understanding” the culture and “getting” it.
Take sport, for example, I’m a massive football (proper football
) fan, and sports fandom is something that translates pretty easily across both cultures. I’ve always known the basic rules of American Football, but never really felt any kind of “connection” to it until moving to New York when now, suddenly, i’ve got a team i can identify with and root for. I’ll happily talk about our performance this season down my local bar - and out of enjoyment of the sport too, not solely in order to feel or look “American.” I “get” football - and over time i’ll also “understand” it as i build up both knowledge of its history and my own experiences.
Conversely, there are also things i will probably “understand” but never “get” purely because the cultural divide is just too great. I understand what the Constitution is, for example, and the role it plays in US politics, government and culture probably better than most americans because its something i’ve actively had to study since being over here in order to get inside the American psyche (and so that i can throw peanuts at the tv like everyone else in the bar when George Bush Jnr is up there talking about how he’s not ignored it
). I’ll never “get” it though. Similarly the Myth of the American Revolution will always be alien to me - largely because i definitely know more about most americans than that (i studied it at university), and because no matter how long i lived here, i’d always be a Red Coat deep down.
I guess what i’m saying, in summary, is that i think it is possible for a non-native to truly comprehend British (or American) things given time, but probably not all of them. The number of those things that will forever remain alien is also probably smaller and the time it takes to grasp the others less lengthy than for many different cultures, however. This is because we do, broadly speaking, share a language and because there are sufficient broadly similar cultural hooks to allow entry into all but the most unique of beliefs (our “first time i went drinking legally” stories and yours may differ by a few years, but they both indubitably ended up with us looking stupid, throwing up and waking up with a road sign and/or traffic cone in our bedrooms).
So basically us foreigners can be just as good Americans as you lot - just remember you’ve probably had a 20 year head start. 
*Business is different of course, business has been sufficiently international for a long time and is sufficiently rules-based for all but the most peculiar cultural differences to stand out.
**i like to call this “Cultural Friction” - the idea that people rely on familiar cultural elements to get them through conversations. The greater the difference in cultural background between you and the rest of the conversations participants, the more you’ll be unable to get a grip on the conversation and eventually slip out of it completely.
***Interestingly i’ve noticed the subtle defensive position that this brings about - there are times i find myself acting more English over here than i do at home out of a need for security in what has become (unwittingly) a hostile cultural environment. Luckily this generally manifests itself in the form of increased tea drinking rather than an overwhelming urge to try and collect back taxes from you and/or burn the Whitehouse.