I don't "feel" American either, but I am absolutely 100% American beef. I once had conceits that I was some sort of cultureless ideal of the sort that American liberals often wish they were. This was proven to be devastatingly naive and comically false by two specific events that occurred about five years ago.

One, I fell for a girl that looked and sounded American but was culturally Arab. She had American parents, grew up in Kuwait, and moved to the US to go to college (one of those people who can honestly claim to be borderless and feel like they don't really fit in in any country). Getting to know her intimately (and I mean personally, not sexually) I began to notice subtle differences in how she behaved that, when I considered them, mapped to very large differences in how we see the world - mine was very much material based in comparison to her, both in the literal (consumerist) and philosophical sense. [as a side note, though, I'm increasingly convinced that women's fascination with shoes and shoe purchasing is biological and not cultural in origin.]

Two, shortly after the topic of point one flamed out, I was laid off from my job. The immediate effect was to realize how much of my identity was tied up in my job, which scared the hell out of me at the time because my belief was that I was beyond that kind of "superficial" identity, and in addition I didn't like the job that much as it was. I was unemployed for about three months and in that time I came to understand why. Part of it was that I clearly enjoyed the material benefits of my job, and the respect (primarily given in non-verbal ways) it earned me from others, often derived from my "material prowess." Closely related was the sense of superiority I garnered from my middle class status (particularly at my young age at the time not long out of school). Since this all operates non-verbally and subconsciously, it's largely unnoticeable until it has been (partially, at least) removed.

Not easy things for a liberal to consider. You can count up the multi-dimensional ironies on your own.

Events like the above don't occur when I'm traveling in Europe or interacting with Europeans. To me the differences feel slight. I'm sure if I were to, say, live in France for a few years and learn the language I'd feel a bit differently, but American culture is derived almost entirely from the Europeans that settled here and the cultures they brought with them. My feelings on that often lead to friction here and elsewhere particularly when I interact with people that don't understand their own relationship with their social class and how it creates them (perhaps not ironically some of the worst offenders are people who rail against class the most). It's also why I laugh a lot when I observe Americans and French (in particular) arguing with each other.

you are the media you consume.