Non-trivial population is sufficient. If there have been several generations of people born into the Gaians- and there are more born into the Gaians than recruited into the Gaians- then while the Gaians will still share a cultural group with Bretonia, they- as a group- are no longer Bretonian in nationality or ethnicity.
In the original post, you did mention "looking and acting like Bretonians"- while no longer "Bretonian" in culture, ethnicity, or nationality, the Gaians have not spent so much time apart from Bretonia that they have formed an entirely separate cultural grouping. They will look, act, and think similarly to Bretonians, but not identically. The difference will not be so large as the gulf between Bretonians and Rheinlanders, and someone unfamiliar with either group might not pick up on the differences immediately, but yes, I agree that there would be subtle differences.