Let's stick with white people.

Some literally don't know. They are probably of German or British (in its various flavors) background but it was centuries ago. Or, their background is mixed. Or maybe they sorta know but literally don't care.

For many people, it's a piece of family trivia, perhaps there are a few holiday traditions or dishes or a vague desire to visit the country of their roots at some point. It is one of their least salient identities (compared to American, or working class, or Catholic, or female, or their university, or their occupation, or their city...). Most Irish-Americans, Jewish Americans, Polish Americans, etc.

For a relatively small minority, it is actually important to them. Perhaps they are only one or two generations from their ancestors who immigrated. Or perhaps they are part of a strong ethic network (businesses, churches, extended families, civic orgs, sports teams, etc.) This is common for Italians and Greeks, where I am from.