A unified Celtic culture never really existed, though the peoples of Western Europe prior to the Roman conquest had some things in common, including an Indo-European language. It was the Greeks, perhaps, and then the Romans who began to classify the people on their frontiers under the handy term Gauls: - said to mean "foreigner". See how cognate all the following terms are: Gaul, Galatia (in Anatolia), Galicia (Spain), Galicia (Poland), Gael (as in Ireland and Scotland), Waloon (Belgium) and Wales.
The languages of Galicia and Asturias are derived from Latin, as is French (though not Breton) as these areas were assimilated into Roman or post-Roman, Christian culture. As such they have nothing in common with the "Celtic" languages which exist on the Atlantic seaboard such as Irish, Scottish or Welsh.
As for ethnicity, again, no "pure" Celtic people ever existed. This is a racist 19th century notion, whereby people were divided up into races. The English were particularly eager to expropriate the term British for themselves and thus classify their troublesome neighbours as "Celts".
Ethnically, if the people of Galicia and Ireland have anything genetic in common, it's likely to be a result of a marauding Viking in the 10th century!
Back to Galego and Portuguese. There's plenty in common, but political divergence - which I believe began when the dioceses of Braga (Portugal) and Santiago split after a row between bishops in the 9th century! - had much to do with the sister languages growing apart as they grew up. But they're still - the sisters that is, not the bishops - on speaking terms, so to speak.