I even had a heated debate about this with a chair of Celtic Studies.
A common language basically only comes about through a shared ethnicity.
There is also a possibility that the word Scoti is a synonym of Celt. *kel-to (āto strike; to fightā) vs *skutÄ, (āto cutā).
There is no surviving evidence for Britons or Gaels calling themselves Celts. However, Caesar states that those dwelling at least in coastal Britain were just Northern Gauls (Celts)
Medieval Irish and Welsh tales also treat mainland Celts as if they are related to them.
The Irish tales speak of the Milesians, who would have been Iberian Celts, having been their ancestors. Irish translations of classical sources clearly understand the Celtic druids and Irish druids have the same origins. Many tales speak of Irish having conquered Europe, aligned with the spread of the La Tene Celts.
The Welsh tales also have brothers Lludd and Llefelys ruling Britain and Gaul. Greco-Roman sources also talk about a relationship between eponymous founders of the Britons and Celts. It was also reported that the druids of Gaul believed they were the descendants of a mixture of Britons & Irish, the native population, and migration from southern Germany.
āIn their own language they are called Celts; in ours, Gauls.ā
~Caesar, De Bello Gallico
This, together with coins inscribed with āCeltitanā found in Spain, and Greek chroniclersā use of the term Keltoi, points towards a shared Celtic identity, regardless of what academics claim.