I guess this can get abstract, when we start talking semantics. Is an atheist simply "without belief" or is "atheism" label strictly "without religion?" I guess the latter would be the common answer, but I don't see it so simply.
I cannot separate religion from fundamental beliefs; they're one in the same to me. What do we call a person who believes in a higher being, but follows no religion?
If by religion, you strictly mean those found in a book, regardless of general beliefs or non-religious spirituality, then I'm more inclined to revise my answer and say: "ethnicity is more important in my life, since I don't believe any strict religion 100% at the moment - thus my answer cannot be religion either way."
I think they're connected, and I can't understand how they cannot be. Consider that "ethnicity" is very loosely defined to begin with, and incredibly inclusive - broadly it is just a common nationality, culture, race, or religion a group shares. Even considering that not all ethnicities are defined by religion (though many are) ethnicity and nationality inevitably influence your religion regardless.
For example, what are the odds that I, and African American would be a Buddhist versus a Christian, considering history and location? Definitely interwoven, without a doubt, just not strictly so.