People believe in all kinds of fanciful nonsense to try to feel "special". In the US in particular, people will draw on some distant real or imagined ancestry to try to establish some kind of feeling of ethnic identity. Part of the reason may be the feeling of vacuousness of American identity from an ethnic point of view, as well as the dissolving religious identity which historically functioned as a substitute for ethnic identity in the US. (Various ideologies and subcultures are also expressions of this.) People will not only claim to belong to ethnicity X, 5+ generations after their ancestors immigrated and 3+ of which didn't speak the language and didn't maintain any contact with the country of origin; they will also claim they're "1/16th" of some ethnicity, as if "genes" or "blood" were like chemical elements. Naturally, these "identities" are rooted in stereotypes rather any kind of living culture.
It's a kind of cosplay-lite for the masses.
I highly recommend reading Ethnic Options by Mary C. Waters. It's a fascinating work of sociology that defines this exact phenomenon and explains its origins.
I'm so glad someone brought this up. It irks me when I hear Americans detail every minor fraction of their genetic makeup: 1/4 Italian, 1/8 German, 1/16... etc. But they don't speak any of these languages, they've never even visited these countries. It's such a matter of pride for a lot of Americans, but it's just a costume.
A quote I found here on HN, that I really liked: "Americans will say they are Italian because their great grandma ate spaghetti once, but God forbid someone is American because he was born there" - mvieira38 (https://news.ycombinator.com/item?id=43930642)