I don't think everyone who speaks a Celtic language is automatically "a Celt". I speak Gaelic and understand a bit of Welsh but I don't consider myself a Celt. I don't think that word has much relevance really these days except as a useful tag to distinguish between language groups or whatever.
I do get really narked at what gets labelled Celtic music though. It's totally overused, I mean if a band has a fiddle player they get called Celtic... eeesh