Historically, the Celts weren’t a single unified group but a collection of tribes sharing linguistic and cultural traits, identified by their Indo-European languages and material culture (like La Tène art). These tribes spread across Europe from around 1200 BCE, with their peak influence during the Iron Age (roughly 500 BCE to the Roman conquests). Archaeological and linguistic evidence places Celtic-speaking peoples in central Europe—like the Hallstatt and La Tène cultures in modern Austria, Switzerland, and southern Germany—and indeed in parts of what’s now Spain (Celtiberians), France (Gauls), and even northern Italy (Cisalpine Gauls). Greek and Roman writers, like Herodotus and Caesar, described these "Keltoi" or "Galli" as widespread, but they didn’t mention Ireland or Scotland explicitly.
So, where do Scotland and Ireland fit? The Celtic languages there—Goidelic (Irish, Scottish Gaelic, Manx) and, to a lesser extent, Brythonic (Welsh, carried by later migrations)—arrived later, likely through a mix of migration and cultural diffusion. Ireland’s earliest inhabitants, pre-1000 BCE, probably weren’t Celtic-speaking; they left megaliths and no written language. The shift to what we call "Celtic" culture in Ireland likely started around 500–300 BCE, possibly from trade or small-scale movements from Britain or the continent, though the exact mechanism’s debated. Linguistic evidence suggests Goidelic languages crystallized there over time. Scotland’s a bit trickier—before the Picts (who may or may not have been Celtic-speaking; their language is a mystery), it’s unclear who dominated. By the Roman era (1st century CE), Celtic languages were taking hold in both places, but they weren’t the "original" inhabitants.
Your point about Spain and France having a stronger early Celtic presence is spot-on. The Celtiberians in Spain mixed with Iberian locals, leaving inscriptions in Celtic languages by 200 BCE, and Gaulish tribes in France were well-documented by Rome before being Latinized. Meanwhile, Ireland and Scotland’s Celtic identity solidified later, post-Roman collapse, as Goidelic and Brythonic speakers entrenched their cultures—think 5th–6th centuries CE, with Christianization and the rise of figures like St. Patrick amplifying the shift.
So why do Americans call Scottish and Irish music "Celtic"? It’s less about ancient tribal accuracy and more about a 19th-century revival. Scholars and nationalists—like the Celtic Revival movement—latched onto the term to connect modern Irish and Scottish traditions to a grand, pre-Roman past, even if the dots don’t perfectly align. The music’s tied to those later Gaelic-speaking cultures, not the continental Celts of Spain or France, whose traditions mostly faded under Roman and Germanic influence. It’s a shorthand that ignores the gaps—like how Pictish or pre-Celtic influences in Scotland might’ve shaped the sound too.
You’re not wrong to call out the disconnect. The "Celtic" label on a jig or a reel is more vibes than strict history—Spain’s Celtic music isn’t on anyone’s playlist today, but a bagpipe tune from the Highlands is.