My husband's family is Irish. None of them were born there, but the name is as about as Irish as you can get and they can trace their roots back to Ireland. I do not know what my heritage is, because I don't know who my bio dad is. On Mom's side, there is some German, English, and maybe some Irish.
We want to buy wedding bands that have the Celtic Knot design or Claddagh on them. I asked a friend who is from County Cork, and he is not offended at all.