I agree with JTT's summary of the Celts generally, but it's not the Basques who are often included in the category. It's the Galicians, from northwestern Spain. Galicia (like much of Europe) once had a Celtic-speaking population, and this can be seen in place names and other linguistic traces. But the language they now speak is basically a Latinate language close to Spanish and Portuguese (referred to as Gallego or Gallego-Portugues). I think it's silly to refer to Galicia as a Celtic province, otherwise most of Europe would be Celtic too.

Most scholars of modern Celtic cultures restrict the meaning of Celtic to a language group. If it refers to people, it refers to those who speak (or whose ancestors in last generation or two spoke) a Celtic language. Archaeologists looking at ancient sites will add certain styles of metalwork, burial practices, etc, to their meanings of Celtic. All scholars recognize that language and cultural forms can be passed among people not genetically related, so no scholar would argue that all speakers of Celtic language share genetic heritage. The DNA studies in this sense are kind of a red herring. BUT, at the same time, there is some probability that many of the people sharing these cultural traits also shared genetic traits, for all the obvious reasons.

As to Celtic Music, it is tempting to say, as Curmudgeon has said, that the idea of Celtic Music originated with record producers who wanted to market music. But this is untrue. People referred to music and culture as Celtic before records existed. In general, the term was first applied by nationalists of the various celtic countries (and their diasporas) making an argument that their cultures were different and valuable, using the word celtic to set them apart from the popular culture around them.

Does Celtic Music exist? Sure! It's a popular music style based on Irish, Scottish, Welsh, etc... folk music. Is English folk music different? Yes, but it's related because there has been a strong mutual influence going back hundreds of years. Is this politically controversial? I don't see why it should be; though of course the nature of the influence is tied to much political controversy, one can't deny the influence itself.