Como dizer "Me identifiquei" em inglês

Se identificar com algo = to be able to relate to something.

Eu me identifico com isso = I can relate to that.

"No qual me identifico muito" = which I can really relate to.

In English they prioritize words that don't seem like Latin or romance languages (when cognates)
So they use "give up" more than "desist", "hate" than "detest" (or - dislike intensely, can't bear, feel aversion, etc.

Identify (with) - I think falls in that category, or almost, but you can see it scattered in writings out there. For instance:
It is well-written and has an interesting plotline, one I can identify with.
That's a perfect textbook example we can use. In this context, we can say the writer of the sentence meant either he/she "relate to" the specific plot of the book or "the plotline is something that inspires him/her" (he reads that kind of plotline - that would be the second type of "identifying himself with.")

As Leonardo pointed out, there are other likely meanings in context: relate to, be in sympathy, put oneself in the shoes of, put oneself in the place of - as in "I put myself in his place, and because I'm familiar with his circumstances.".
It could also be that something "...strikes or touches a chord/hits home, etc.

Para situações não relacionada a pessoas, como por exemplo, "Logo me identifiquei com a cultura dos EUA", "Me identifiquei com essa música", também se utiliza essa regra?

It's not a rule set in stone, but yes, normally you can use it. In the case of the given sentences, I would say that "I soon identified myself with the American culture." which would be in the sense of "I soon identified myself with the ideas, ideals, or the mores of that culture." It would be in a more general sense.

In that case could be also in the sense of "I relate to American culture" in a given situation, say if you are discussing a scene of a movie in which a guy feels uncomfortable with someone getting too close (and you don't like people getting into your personal space. That is, it's a specific situation you can relate to.

Similarly, but with a subtle difference is the case of the music sentence:
I relate to that music (it evoke memories, triggers emotions, feelings, felt at the time).

That kind of music is something I identify myself with. (that is my thing, I am associated with, that is what I like to listen, etc).

Although sometimes both nuances may (or may not) intersect, Amy Winehouse said in an interview:
And I have plenty of exes, so I can immediately identify myself with that song. ... You have to be thrifty with that kind of music.

That is, that song she referred to evokes memories, and is something she sings, something she is into.