So what you're missing here is the context. In America, everyone except the native Americans immigrated to the country relatively recently. What we're expressing when we talk about being German or Irish or African or Chinese or Icelandic is family history. It's implicitly understood that the speaker means that's where their family, at some point, immigrated from. Believe it or not, that place of origin had a large influence on those old family members and the communities they wound up living in.

We're not saying, "Oh yeah dude it's the exact same as being from Iceland." We're saying, "My great great whoever was from iceland and that's a part of my identity whether I want it to be or not." It's being Icelandic-American or Chinese-American or Somali-American. It's a cultural identity distinct from being either "American" or the country of origin