Ethnically, the Spanish Basques and the French Basques are identical.Culturally, they differ not because of their nationalities, but because of the level of assimilation over the last 200 years.
In Spain, except for the period of Generalisimo Franco, the Basques have always enjoyed a high level of automony.This tradition of self-rule goes deep into the Middle Ages.
In France, the Basques (as well as all the other ethnic minorities prevalent in the 18th Century) were absorbed by the centralizing forces of the French Revolution.Most important were the educational reforms implemented by Emperor Napoleon Bonaparte that led to the creation of the French national identity.That is why today's French Basques are culturally distinct from Spanish Basques.
In terms of the Basque language, you are almost correct.The Basque language, Euskera, predates the Indo-European root of all European, Indian, Persian, etc., languages.I don't know how many thousands of years that is, but clearly, it is older than the Mesopotamian origins of Western Civilization.
The Basques themselves like to joke (half seriously) that they are the descendants of the Cro-Magnon who painted the walls and ceilings of the prehistoric caves in Spain and France.It's a plausible theory, but we'll never know.