Can someone identify as just one of their ethnic backgrounds, though they have multiple ones?

Identity, and indeed ethnicity, are not about blood or DNA or ancestry. Yes, skin colour, as a “visible minority”, is different, particularly in the States; but that’s not the scenario your question puts forward. And even in that case, identity is not about blood. It’s about what your society makes of what you look like.

Ethnicity is a cultural construct. Particularly when people around you can’t tell, just by looking at you, which I assume is the case with you. (Disclosure: as you can tell from the topics, one of OP’s ethnicities is Greek, and I’ve had exchanges on that with him before.)

If you prioritise one ethnic background over another, noone has the right to stop you, go ahead: that’s how you identify. To take a smaller-scale example, I identify with my mother’s region of origin rather than my father’s: I’ve spent four years living in Crete, a month visiting Cyprus; I use Cretan dialect words, with only a very faint Cypriot accent; I’m not going to identify with something I’ve barely experienced.

The catch with identity is that identity involves a community, not just an individual; and the community you identify with may not identify with you. Diasporans routinely get a rude shock when they go back to the home country, and the people in the home country say “Huh? Nah, You American.” I’ve seen that here on Quora from Iranians, Italians, and Greeks; it was visibly default behaviour in Armenia.