It mostly depends on how the culture you want to join perceives identity

I disagree -- humans, in particular, adults, are not that malleable. Discarding your old identity is hard.

Of course, some cultures are more accepting of newcomers (e.g. US) and some less (e.g. Japan).

it's easier to become American than to become Jewish.

I think of "Jewish" as mostly ethnicity (if you prefer, a particular gene pool) and somewhat culture. In that sense you cannot "become" Jewish. You probably mean "convert to Judaism", though, and that's not that hard to do. Judaism does not proselytize for historical reasons, but if you want to convert you can do so.